N-Dimensional Cross Product
Robert J. Swartz

We are concerned with the multiplication of n-vectors using the cross product.  For example, the 3-D cross product is defined as follows:

A x B =det|x1, y1, z1|    where A=[x1, y1, z1] and B=[x2, y2, z2]
|x2, y2, z2|
|i,     j,    k|

The question is , how do we multiply vectors of arbitrary dimensions.  For example, in 4 dimensions, we have the following:
x(A, B, C)=
det|x1, y1, z1, w1|
|x2, y2, z2, w2|
|x3, y3, z3, w3|
|i,    j,    k,   l   |

In 5 dimensions, for example, we have

x(A, B, C, D)=
det|x1, y1, z1, w1, t1|
|x2, y2, z2, w2, t2|
|x3, y3, z3, w3, t3|
|x4, y4, z4, w4, t4|
|i,    j,    k,    l,   m|

As we can see, the 3-D cross product is a binary operation, the 4-D cross product is ternary, and the 5-D cross product is quatenary.
These operations are not commutative.  For example,
x(A, B, C)=-x(A, C, B)
Proof:
det
|x1, y1, z1, w1|      det|x1, y1, z1, w1|

|x3, y3, z3, w3|            |x2, y2, z2, w2|
|x2, y2, z2, w2|   =  -  |x3, y3, z3, w3|
|i,    j,    k,   l   |

|i,    j,    k,   l   |