Let $\mathcal{F}$ be a locally free sheaf of rank $d$ on a scheme $X$ together with an epimorphism $\mathcal{O}_X^n \to \mathcal{F}$. Now due to abstract reasons (Plücker embedding, Serre's results on coherent sheaves etc.) there is a canonical exact sequence of the form $(\wedge^d \mathcal{F})^{\otimes k_2})^{r_2} \to (\wedge^d \mathcal{F})^{\otimes k_1})^{r_1} \to \mathcal{F}^* \to 0$. Here $\mathcal{F}^*$ is the dual of $\mathcal{F}$. Canonical means that the left morphism is a matrix which is built up out of exterior powers and tensor products of the given global sections of $\mathcal{F}$, in a way that does not depend on any other choices. But how can we get this sequence explicitly? I already know that there is a canonical exact sequence ${(\wedge^d \mathcal{F})^{\*}}^{\binom{n}{d+1}} \to \mathcal{O}_X^n \to \mathcal{F} \to 0$. The naive idea just to dualize it does not work since the arrows reverse their direction.