Given a smooth complex projective variety $X$, I want to try and learn how to find a locally free resolution $$ 0 \to \mathcal{O}_X^\vee \to \mathcal{L}_0 \to \mathcal{L}_1 \to \cdots $$ The starting cases I'm interested in are smooth hypersurfaces and smooth complete intersections, but I hope to be able to extend this to more cases. Moreover, I am really only interested in finding the first two terms $\mathcal{L}_0,\mathcal{L}_1$ since I only plan on using this resolution for computing $\mathcal{Ext}^i(\mathcal{E},\mathcal{O}_X)$.
On second thought, this question is kind of pointless. If I can find a locally free resolution $$ \mathcal{L}_\bullet \to \mathcal{E} \to 0 $$ then computing $\mathcal{Ext}^i(\mathcal{E},\mathcal{F})$ is easy, so long as the locally free sheaves come from $\mathbb{P}^n$. This is because $$ \mathcal{Hom}(\mathcal{O}(a),\mathcal{F}) \cong \mathcal{F}(-a) $$ and we have $\mathcal{Hom}$ playing nice with respect to finite direct sums in the first factor.