Given a connection \(\nabla\) on \(E\searrow M\) a Riemannian manifold the curvature form is a 2-form \(F_\nabla\) with values in \(\mathop{\mathrm{End}}(E) \cong E {}^{ \vee }\otimes E\), i.e. \begin{align*} F_\nabla \in \Omega^2_{M}(\mathop{\mathrm{End}}E) \cong {{\Gamma}\qty{\bigwedge\nolimits^2 {{\mathbf{T}}M} \otimes\mathop{\mathrm{End}}E} } \qquad F_\nabla(X, Y)({-}) = \qty{ [\nabla_X, \nabla_Y] -\nabla_{[X, Y]}} ({-}) \end{align*}

Every smooth manifold admits a Riemannian metric, so consider Riemannian manifold.