Multilinear form as scalar multiple of determinant function

Question

While going through Hungerford's $\textit{Algebra}$, there is a theorem of linear algebra which states that every alternating $R$-multilinear form $f$ on $M_n(R), R$ a commutative ring, is a unique scalar multiple of the determinant function $d:M_n(R) \rightarrow R$.

Now in the proof sketch of this theorem, Hungerford lets $f(I_n)= r \in R$ with $d$ the determinant function. Then he suggests to verify that $rd: M_n(R) \rightarrow R$ given by $$A \mapsto r|A|= rd(A)$$ is also an alternating $R$-multilinear form on $M_n(R)$ so that $rd(I_n)= r \cdot 1= r$. Is there any quicker way to verify this without having to use permutations?

Answer 1

Hint (maybe): what if ones mimics all the way the vectorspace's Grassmann Algebra (Exterior Algebra) construction.