Hi. I was investigating through this week why there are the differential forms, why are they anti-symmetric, why do we have the Jacobian when expressing the volume in a different coordinate system. This was just fantastic! I found all the connections between these topics. And I found that all of these things are due to the fact that the volume element (the same works for an area element) is defined through what is called the triple product, namely ##(A \times B) \cdot C## for vectors ##A, B## and ##C##. But now I wonder: why does the vector product is defined in terms of a determinant in first place? I thought at first that it had to do with the way basis vectors transform between coordinate systems, but I noticed that it doesn't matter, because covectors are defined in such a way that the inner product is left invariant anyway. So why the vector product is defined that way?