No. In n dimensions a cross product of n-1 n-vectors is the determinant with the top row being the basis elements (e_1, e_2, ..., e_n) and the next n-1 rows being the n-1 vectors. The result is a n-vector orthogonal to the n-1 vectors.

In 2 dimensions it is i j x y

In 3 dimensions it is

i j k x1 y1 z1 x2 y2 z2

And so on.