Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated.
When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong R^{\oplus {d\choose n}}, \end{equation} and there is a very canonical isomorphism which maps \begin{equation} (\Sigma r_{j1}e_j)\wedge(\Sigma r_{j2}e_j)\wedge\cdots\wedge(\Sigma r_{jn}e_j) \end{equation} to the determinants of ${d\choose n}$ minors of the matrix $(r_{ji})$.
For general modules things might not be so neat. I still wonder, however, whether we have something similar that can help me understand exterior powers more concretely and make computations more direct.
Thanks very much!