$$\gdef\xto\xrightarrow \gdef\End{\z{ End}}$$

Let $E$ be a vector bundle over a smooth manifold. The connection $\nabla$ is given as such $$ \Omega^0(M, E) \xto{ \nabla} \Omega^1(M, E) \xto{ \nabla} \Omega^2(M, E) \cdots. $$ And the curvature is defined succinctly as $$ F_\nabla = \nabla^2 $$ which is a map $\Omega^k(M,E) \to \Omega^{k+2}(M,E)$ that commute with $C^\infty(M)$ action, hence $F_\nabla \in \Omega^2(M, \End(E)).$

Locally, if we have a base coordinates $x_i$, and local trivialization $e_\alpha$ of $E$, then we may write the connection as $$ \nabla = d + \Gamma_i^\alpha_\beta dx^i \otimes e_\alpha \otimes \delta^\beta $$ where $\{\delta^\alpha\}$ is the dual basis to $\{e_\alpha\}$, and $e_\alpha \otimes \delta^\beta$ is a local section of $\End(E)$. $$ F_\nabla = \nabla^2 = (F_{ij})^\alpha_\beta dx^j \wedge d x^i \otimes e_\alpha \otimes \delta^\beta$$ where $$ (F_{ij})^\alpha_\beta = \d_i \Gamma_j^\alpha_\beta - \d_j \Gamma_i^\alpha_\beta + \Gamma_i^\alpha_\gamma \Gamma_j^\gamma_\beta - \Gamma_j^\alpha_\gamma \Gamma_i^\gamma_\beta $$