The cross product does exists in any number or dimensions (at least 2). In n dimensions, the cross product of n-1 vectors is the determinant with the axis vecto

The cross product does exists in any number or dimensions (at least 2). In n dimensions, the cross product of n-1 vectors is the determinant with the axis vectors as the first row and each of the n-1 vectors as the next rows. In 3 dimensions, n=3, so the first row is (i, j, k) and the next 2 rows are the two vectors.