The cross product and scalar triple product generalize to any number of dimensions (probably with different names). CrossN takes N-1 arguments and permuting the arguments has the same effect as the triple scalar product (even is equal, odd is negative). The ScalarProductN would take N parameters and relates to permutations and determinants in the same way.

Example “useful” feature, a 4d plane (N,-d) equation is the “cross” of the homogeneous versions of 3 input points. Obviously there’s a cheaper way to compute it than a generic cross in this specific circumstance. Inverses can also be computed in a similar way because of the relationship between cross and the null space of given set of vectors. Specifically, your dual basis a’b'c’ put in matrix form is the matrix inverse of abc.

An interesting possible extension was http://www.geometrictools.com/Documentation/LaplaceExpansionTheorem.pdf . It kind of looks similar to cross, but I don’t know if it’s just coincidence yet. That specific formulation leads to a decent, intuitively vectorized 4×4 matrix inverse at least.