Given a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endom

Given a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endomorphism of a 1-dimensional vector space is a number. This number is the determinant. This is probably the correct generalization of the volume definition.