There are two kinds of generalizations of the cross product to higher dimensions: the perpendicular product and the bivector product.

The bivector product can be rationalized using the geometric product, given any arbitrary pair of vectors in N dimensions, it produces a bivector which is essentially an oriented 2D plane with associated magnitude.

The perpendicular product generalizes the (pseudo-)determinant definition of the cross product, i.e., as the "determinant" of the following pseudo matrix:

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`[ x1 y1 z1 ]`

[ x2 y2 z2 ]

[ X Y Z ]

corresponding to the cross product of <x1,y1,z1> with <x2,y2,z2>. X, Y, Z are the basis vectors of 3D space; when you expand this determinant and group the terms by X, Y, Z, you get the components of the cross product of the two vectors. The N dimensional analogue, therefore, is an NxN matrix with the basis vectors of N-space as the bottom row, and the components of (N-1) vectors filling the rest of the rows. So it's a product not of 2 vectors, which only occurs in 3D, but of (N-1) vectors. So a 4D perpendicular product is a ternary operator taking 3 vectors, a 5D perpendicular product is a quatenary operator taking 4 vectors, etc..

For a generalization of electromagnetism, I'd hazard to guess that the bivector generalization is probably more useful, since I wouldn't know how to rewrite the Maxwell equations for 4D if I had to add an extra vector into every cross product (where would that vector come from?). It also "makes more sense" physically-speaking: if you have 2 interacting charged particles, it would be strange if a 3rd vector appeared out of nowhere just to satisfy the requirements of the mathematical perpendicular product. That said, having an extra dimension probably has some fundamental consequences as to how electromagnetism would behave in 4D. In 3D a circulating electric current induces a directed magnetic force parallel to the "extra" dimension; in 4D there would be two dimensions leftover, so what would the magnetic field be? A circulating field in the orthogonal plane? In 3D, a linearly-travelling electric charge induces a circulating magnetic field around the line; in 4D there would be 3 dimensions left over. What shape would the magnetic field take? Whatever the answers, it will certainly produce some fundamentally different behaviours than in 3D.