A coordinate-free view on "the" generalized cross product

The higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This coordinate-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann's, Jacobi's and Room's identities. Moreover, such a view provides a the higher dimensional analogue of the decomposition of the vector Laplacian which itself gives an explicit coordinate-free Helmholtz decomposition in arbitrary dimensions n ≥ 2.