Given a cross product in n dimensions, it's possible to construct a normed division algebra in n+1 dimensions. Hurwitz's theorem says that the only normed divis

Given a cross product in n dimensions, it's possible to construct a normed division algebra in n+1 dimensions. Hurwitz's theorem says that the only normed division algebras are the reals, complexes, quaternions and octonions. So there can be no cross products outside of 0, 1, 3 and 7 dimensions. In computing applications, if you want to find a vector perpendicular to two or more vectors outside of 3 and 7 dimensions, you would use the QR decomposition to do it. If on the other hand, you want to generalise the "directed area" properties of the cross product, you would use Exterior Algebra, or its elaborations like Clifford Algebra or the theory of differential forms.