Let $M$ be an $n$-manifold. Let $E_1, E_2,\dots, E_n : U\subset M \to TM $ be a local frame for $TM$ with associated local dual frame $\epsilon^1, \epsilon^2,\dots, \epsilon^n : U\subset M \to T^*M $.
Suppose that $[E_i, E_j] = \sum_{k=1}^n c_{ij}^k E_k$ the exercise asks to show that $d \epsilon^k = - \sum_{i,j} c_{i j}^k\ \epsilon^i \wedge \epsilon^j = \sum_{i<j} (c_{j i}^k - c_{i j}^k) \ \epsilon^i \wedge \epsilon^j $.
I've tried to solve it in the following way: $d \epsilon^k (E_i, E_j) = E_i (\epsilon^k(E_j)) - E_j (\epsilon^k(E_i)) - \epsilon^k([E_i, E_j])$ since $E_i (\epsilon^k(E_j)) = 0$ for any $i,j,k$ I've obtained that $d \epsilon^k (E_i, E_j) = - \epsilon^k([E_i, E_j]) = -c_{i j}^k$, thus $d \epsilon^k = \sum_{i<j} -c_{i j}^k\ \epsilon^i \wedge \epsilon^j $.
Where is the mistake? It is possible that the exercise is not correct? How can it be corrected?.