Let $M$ be a smooth manifold, $(E_i)$ a smooth local frame for $M$, and ($\varepsilon^i$) the dual coframe. For each $i$, let $b^{i}_{jk}$ denote the component functions of the exterior derivative of ($\varepsilon^i$), i.e, $$d\varepsilon^i = \sum_{j<k} b^{i}_{jk} \varepsilon^{j} \wedge \varepsilon^{k},$$ and for each $j$, $k$, let $c^{i}_{jk}$ be the component functions of the Lie bracket $[E_j, E_k]$, i.e, $$[E_j, E_k] = c^{i}_{jk}E_i.$$ I wanna to prove that $b^{i}_{jk} = -c^{i}_{jk}$.
Using the formula $d\omega(X, Y) = X(\omega(Y)) - Y(\omega(X)) - \omega([X, Y])$, I know that $$d \varepsilon^{i}(E_u, E_v) = - \varepsilon^{i}([E_u, E_v]) = - c^{i}_{uv},$$ but I can't see how to finish. How to prove that, on the other hand, $d\varepsilon^{i}(E_u, E_v) = \displaystyle\sum_{j<k} b^{i}_{jk} \varepsilon^{j} \wedge \varepsilon^{k}(E_u, E_v) = b^{i}_{uv}?$