Two vectors have different angles and magnitudes. When you combine these vectors you expect the combination to be a sort of 'average' of the the angle between the two original vectors, and you expect the magnitude to increase/decrease according to the magnitudes of the originals.

When you multiply two vectors using the cross product you are doing the same thing. The only real difference being the inclusion of the unit vector n, which merely specifies that the cross product of two vectors is normal to both.