[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.