I am studying an introduction to differential forms, where an m-form on $T_pR^n$ is a function $\omega$ from $T_pR^n$ to $R$ such that $\omega$ is multilinear and alternating.
I understand how to show the determinant is a multilinear and alternating map, but how would one show the determinant is the unique multilinear and alternating map? (i.e. so these two algebraic conditions force $\omega$ to act as a determinant).
Thank you for any ideas/help!
NOTE: With the added property that $det(I)=1$