This is Vakil 13.5 G, self-study. I have already spent three hours trying to solve the problem and research the answer, and would consequently like a complete answer. I have seen 3 different answers in three different solution guides to Hartshorne's comparable exercise 2.5.16, and all of them were seemingly different, and all claimed everything was obvious.
We are given a locally free sheaf $\mathcal F$ of rank $n$, and asked to describe a map
$$\bigwedge\nolimits^r \mathcal F \times \bigwedge\nolimits^{n - r} \mathcal F \to \bigwedge\nolimits^n \mathcal F$$
inducing an isomorphism
$$\bigwedge\nolimits^r \mathcal F \to \Big(\bigwedge\nolimits^{n - r} \mathcal F\Big)^* \otimes \bigwedge\nolimits^n \mathcal F$$
This topic really seems to assume you have taken Differential Geometry before, which I have not. If I had to guess, I would say the map on an open set on which $\mathcal F$ is free is
$$(e_1 \wedge ... \wedge e_r, e_{r + 1}\wedge ... \wedge e_n) \mapsto (e_1 \wedge ... \wedge e_r \wedge e_{r + 1}\wedge ... \wedge e_n)$$
if the $e_i$ form a basis for $\mathcal F$ locally, but I am not actually sure what I am really saying here. I feel I am just writing down expressions that look correct and not truly grasping what is happening.
Even if my "guess" is correct, I have no idea how this induces the isomorphism asked of us.