The second term is not a cross product. It's the exterior product (or bivector). Cross product only works in 3D. Exterior product can work in any higher dimension.
A cross product of two 3D vectors is another vector, perpendicular to the plane containing the two vectors. The exterior product is a 2-vector (thus bivector) that sweeps the parallelogram between the two vectors; it's on the plane containing the two vectors. In 3D, the cross product vector is perpendicular to bivector plane.
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.
The exterior product is the "correct" version of the cross product.
I'm OK with the good guys seizing the name.
Cross product works in dimension 2^k-1 for k in 0,1,2,3. (real / singleton, complex / duonion, quaternion, octonion)
It is trivial in 0D and 1D, defined uniquely in 3D, and unique up to some arbitrary negations in 7D.
A cross product of two 3D vectors is another vector, perpendicular to the plane containing the two vectors. The exterior product is a 2-vector (thus bivector) that sweeps the parallelogram between the two vectors; it's on the plane containing the two vectors. In 3D, the cross product vector is perpendicular to bivector plane.