I find both the geometric and algebraic definitions quoted here unsatisfying. What is a “volume” spanned by a vector space of polynomials or co-tangent function

I find both the geometric and algebraic definitions quoted here unsatisfying. What is a “volume” spanned by a vector space of polynomials or co-tangent functionals? *A* determinate function (not the) is simply a skew symmetric n-linear map into the underlying field. Done. Now we get the volume interpretation when it’s appropriate, the wedge product interpretation, and the generalization to finitely generated projective modules (if a determinate function exists, there are additional conditions needed for the existence.)