Using the orthonormal basis V = { V_{0}, V_{1}, V_{2}, V_{3}, V_{4}, V_{5}, V_{6} }, here is the baseline 2in7 cross product, chosen arbitrarily from 480 possibilities:
… × V_{0}  … × V_{1}  … × V_{2}  … × V_{3}  … × V_{4}  … × V_{5}  … × V_{6}  
V_{0} × …  0  +V_{2}  −V_{1}  +V_{4}  −V_{3}  +V_{6}  −V_{5} 
V_{1} × …  −V_{2}  0  +V_{0}  +V_{5}  −V_{6}  −V_{3}  +V_{4} 
V_{2} × …  +V_{1}  −V_{0}  0  −V_{6}  −V_{5}  +V_{4}  +V_{3} 
V_{3} × …  −V_{4}  −V_{5}  +V_{6}  0  +V_{0}  +V_{1}  −V_{2} 
V_{4} × …  +V_{3}  +V_{6}  +V_{5}  −V_{0}  0  −V_{2}  −V_{1} 
V_{5} × …  −V_{6}  +V_{3}  −V_{4}  −V_{1}  +V_{2}  0  +V_{0} 
V_{6} × …  +V_{5}  −V_{4}  −V_{3}  +V_{2}  +V_{1}  −V_{0}  0 
The other 479 differ only by permutation of basis elements and changes of sign.
With V_{n} corresponding to U_{n+1}, there is a clear resemblance to this excerpt from the baseline 3in8 table:
… × U_{1}  … × U_{2}  … × U_{3}  … × U_{4}  … × U_{5}  … × U_{6}  … × U_{7}  
U_{0} × U_{1} × …  0  +U_{3}  −U_{2}  +U_{5}  −U_{4}  +U_{7}  −U_{6} 
U_{0} × U_{2} × …  −U_{3}  0  +U_{1}  +U_{6}  −U_{7}  −U_{4}  +U_{5} 
U_{0} × U_{3} × …  +U_{2}  −U_{1}  0  −U_{7}  −U_{6}  +U_{5}  +U_{4} 
U_{0} × U_{4} × …  −U_{5}  −U_{6}  +U_{7}  0  +U_{1}  +U_{2}  −U_{3} 
U_{0} × U_{5} × …  +U_{4}  +U_{7}  +U_{6}  −U_{1}  0  −U_{3}  −U_{2} 
U_{0} × U_{6} × …  −U_{7}  +U_{4}  −U_{5}  −U_{2}  +U_{3}  0  +U_{1} 
U_{0} × U_{7} × …  +U_{6}  −U_{5}  −U_{4}  +U_{3}  +U_{2}  −U_{1}  0 
Let W = { W_{0}, W_{1}, W_{2} … W_{n−1} } be an orthonormal basis of an ndimensional vector space over the real numbers. Then there is a widely recognized definition of the cross product of n − 1 factors, and it is most succinctly defined as the determinant of a formal matrix.
Here it is for the case of three vectors in four dimensions (the "3in4"):
A × B × C = det 

The extension to other numbers of dimensions is so obvious that there is little need to try writing the general formula. The twofactor threedimensional cross product, indispensible to physicists, looks like this:
A × B = det 

With four factors in five dimensions, the following is obtained:
A × B × C × D = det 

Aside from the 2in7 and 3in8, mathematicians rarely define any cross product that does not fall into this determinantbased pattern. However, the next section describes a different multiplication, using multiple vector spaces, that resembles the cross product of n − 1 factors in n dimensions.
In all the discussions so far, the assumption has been that the output of cross multiplication resides in the same vector space as the inputs. If this constraint is lifted, another kind of generalization of the cross product becomes possible, namely the wedge product. Although the wedge product is often defined abstractly by certain of its behaviors, in this report we give a more concrete presentation by way of example.
Start with a vector space G_{1} that has orthonormal basis { e_{0}, e_{1}, e_{2}, e_{3} }; it helps to call vectors within G_{1} univectors. For any two vectors A and B in G_{1}, the wedge product is symbolized A ∧ B. The wedge product, like most multiplications, is distributive over addition:
A ∧ (B + C) = (A ∧ B) + (A ∧ C)
Like cross products, the product of two vectors from G_{1} is anticommutative:
A ∧ B = − B ∧ A
A ∧ A = 0
and it is linear in each factor:
(xA) ∧ (yB) = xy(A ∧ B)
Unlike the cross products, the wedge product is associative:
(A ∧ B) ∧ C = A ∧ (B ∧ C)
Thus with basis vectors:
e_{i} ∧ e_{j} = − e_{j} ∧ e_{i}
e_{i} ∧ e_{i} = 0
A key feature of the wedge product is that although e_{i} ∧ e_{j} is a valid expression, it cannot be simplified. In this regard, it resembles an expression like 3 + 4i in complex numbers. Moreover, e_{i} ∧ e_{j} is a vector that resides in the sixdimensional space G_{2}:
an orthonormal basis for G_{2} = { e_{0} ∧ e_{1}, e_{0} ∧ e_{2}, e_{0} ∧ e_{3}, e_{1} ∧ e_{2}, e_{1} ∧ e_{3}, e_{2} ∧ e_{3} }
In building the basis, it little matters whether we choose e_{0} ∧ e_{1} or its negative e_{1} ∧ e_{0}; the same vector space results either way. A condensed notation for the products of G_{1}'s basis vectors is convenient: e_{ij} = e_{i} ∧ e_{j}. Of course, e_{ij} = − e_{ji}. Thus:
the same basis for G_{2} = { e_{01}, e_{02}, e_{03}, e_{12}, e_{13}, e_{23} }
An element of G_{2} is called a bivector.
Multiplication is not limited to two vectors. Consider that e_{i} ∧ e_{j} ∧ e_{k} (more briefly e_{ijk}) is nonzero when i, j and k are distinct. Such a product of three univectors resides in the fourdimensional space G_{3}:
an orthonormal basis for G_{3} = { e_{012}, e_{013}, e_{023}, e_{123} }
Because of associativity, there is little substantive difference between the product of three univectors versus the product of one univector and one bivector. Continuing the pattern of names, elements of G_{3} are trivectors.
Last in the sequence is the onedimensional G_{4}:
a basis for G_{4} = { e_{0123} }
While vectors in this space might be called quadrivectors, they are more likely termed pseudoscalars, because G_{4} is isomorphic to scalar space, which itself could be labeled G_{0} and which would have the basis { 1 }.
In total, there are five disjoint vector spaces here: G_{0}, G_{1}, G_{2}, G_{3}, and G_{4}. With vector A in G_{a} and vector B in G_{b}, a general result is that A ∧ B, if not zero, is in G_{a+b}.
In the threedimensional case, a likely choice for the vector spaces would be:
space  an orthonormal basis  containing 

F_{0}  { 1 }  scalars 
F_{1}  { e_{0}, e_{1}, e_{2} }  univectors 
F_{2}  { e_{12}, e_{20}, e_{01} }  bivectors 
F_{3}  { e_{012} }  trivectors or pseudoscalars 
Using the Hodge dual, we can consolidate four vector spaces into two by declaring that:
In a consolidation of the earlier example, G_{4} would be identified with G_{0}, and G_{3} would be identified with G_{1}, but G_{2} would stand alone.
This consolidation is not always appropriate. Suppose the components of an F_{1} vector are not merely numbers, but numbers carrying a physical unit, say meters. Then the components of an F_{2} vector would be numbers carrying the physical unit square meters, and it is very difficult to establish a plausible identity between meters and square meters.