This is a question of what it "means". It's origins can be traced to problems dealing with SO(n), and in this sense it's uniquely defined by the Lie-algebra of \mathfrak{so}(n), but in the Clifford-algebra world they become blades (more appropriate for CG).

I consider the cross-product to be one of the most abused objects in Physics; my first course in Newtonian mechanics was a nightmare, filled as it were, with the ghosts of all the characters killed so as to make the plot ostensibly "simple".