Let $M$ be a smooth manifold, let $E\to M$ be a smooth complex vector bundle of rank $r$, and let $\nabla$ be a smooth complex-linear connection on $E$ with curvature
For each $k\in\{0,\dots,r\}$, define $c_k(E,\nabla)\in \Omega^{2k}(M;\mathbb C)$ by the Chern-Weil evaluation of the coefficient polynomial $p_k:\mathfrak{gl}(r,\mathbb C)\to\mathbb C$ determined by
in the standard Chern-Weil convention, so in particular $c_0(E,\nabla)=1$. Then each $c_k(E,\nabla)$ is closed. Moreover, for every $k\in\{0,\dots,r\}$, the de Rham cohomology class
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\begin{align*}
c_k(E):=[c_k(E,\nabla)]\in H^{2k}_{\mathrm{dR}}(M;\mathbb C)
\end{align*}
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is independent of the chosen smooth complex-linear connection $\nabla$.