All versions of the cross-product are vector-representations of bi-vectors. The 7d cross-product exists, but in general for [for a,b in R^7] a x b does not have

All versions of the cross-product are vector-representations of bi-vectors. The 7d cross-product exists, but in general for [for a,b in R^7] a x b does not have a unique solution, so it's pretty unsatisfactory as an extension. The problem is, the cross-product uses the normal vector of a plane to represent a 2-d plane (the unique plane that a and b lie on). But 2-d planes only have a unique normal vector in 3-dimensions. With more dimensions, you have more normal vectors, and more answers to a x b.