Determinantal function ($D(I_n)=1$)

A determinantal function on $M_n(F)$ is an alternating multilinear map of the $n$ columns, normalized to take value one at the identity matrix. It is uniquely the determinant. Indeed, expand each column in the standard basis. Terms with a repeated basis vector vanish, and all remaining terms differ from the value on the standard basis only by the sign of a permutation. Thus an alternating multilinear map of $n$ columns is determined by its value on the identity matrix.