[mjcohen]

No. In n dimensions a cross product of n-1 n-vectors is the determinant with the top row being the basis elements (e_1, e_2, ..., e_n) and the next n-1 rows being the n-1 vectors. The result is a n-vector orthogonal to the n-1 vectors.

In 2 dimensions it is i j x y

In 3 dimensions it is

i j k x1 y1 z1 x2 y2 z2

And so on.

[itishappy]

That's not a particularly fruitful generalization. The traditional definition begins:

> The cross product on a Euclidean space V is a bilinear map from V × V to V.

This is required for some of the cross product's useful properties such as:

  |a x b| = |a| |b| sin(theta)

This only works in 0, 1, 3, and 7 dimensions.