I'm having trouble understanding the relation between these two cohomology theories. I know that for a constant sheaf, the sheaf cohomology is essentially singular cohomology and by the de Rham isomorphism is isomorphic to de Rham cohomology. But I'm having trouble understanding the further extent to which they agree (if at all).
In particular, I'm having trouble understanding the following: given a ses of holomorphic vector bundles $$0 \rightarrow \cal{F} \rightarrow \cal{E} \rightarrow \cal{G} \rightarrow 0 $$ any unitary connection $A$ on $\cal{E}$ has, by the Gauss-Codazzi equations, the following shape $$\begin{pmatrix} A_\cal{F} &\beta \\ -\beta^* & A_{\cal{G}}\end{pmatrix}$$ and allegedly $\beta$ represents the cohomology class $\operatorname{Ext}^1(\mathcal{G}, \mathcal{F}) = H^1(\cal{G}^\vee \otimes \cal{F})$. However, I don't understand what the isomorphism between $H^1(\cal{G}^\vee\otimes \cal{F})$ and $H^1_{\text{dR}}\otimes\operatorname{Hom}(\cal{G}, \cal{F})$ looks like, since I don't see any obvious way the de Rham isomorphism carries through. Any help would be appreciated!