Koszul complex and locally free resolution

Question

Let $V$ be an $n$-dimenational vector space. We consider the tautological sequence on the Grassmannain $Gr_{k}(V)$ $$ 0 \to \Gamma \to V \times Gr_k(V) \to Q \to 0,$$ and the projection $p:V \times Gr_{k}(V) \to Gr_{k}(V)$. How can I prove that the Koszul complex $$ \mathcal{K}_{\bullet}(Q^*): 0 \to \bigwedge^{n-k}(p^*Q^*) \to \cdots \to \bigwedge^2(P^*Q^*) \to \mathcal{O}_{V \times Gr_{k}(V)} $$ is a locally free resolution of the structure sheaf $\mathcal{O}_\Gamma$ as $\mathcal{O}_{V \times Gr_{k}(V)}$-module? Can you give me the foundamental steps?