Consider two coherent sheaves $\mathcal{E},\mathcal{F}$, with locally free resolutions \begin{align} &0 \longrightarrow \mathcal{E}_1 \longrightarrow \cdots \longrightarrow \mathcal{E}_n \longrightarrow \mathcal{E} \longrightarrow 0 \\ &0 \longrightarrow \mathcal{F}_1 \longrightarrow \cdots \longrightarrow \mathcal{F}_n \longrightarrow \mathcal{F} \longrightarrow 0. \end{align} The Ext-groups can be computed via the spectral sequence \begin{align} E_2^{pq} \equiv H^p(X, \mathcal{Ext}^q(\mathcal{E}, \mathcal{F})) \Rightarrow \mathrm{Ext}^{p+q}(\mathcal{E},\mathcal{F}), \end{align} and in turn the $\mathcal{Ext}$-sheaves can be related to the locally free resolution of$\mathcal{E}$ by \begin{align} \mathcal{Ext}^i (\mathcal{E}, \mathcal{F}) = H^i( \mathcal{Hom}(\mathcal{E}_\bullet, \mathcal{F}). \end{align} My question is: Is there any way, I can also express the last equation in terms of the locally free resolution of $\mathcal{F}$ at the same time?
Thanks in advance!