Answer by Ryan Unger for In the orthonormal tetrad method, what is the relation between basis one forms and commutation coefficients?

I will use the notation $\theta^i$ for the dual basis since $\omega$ is reserved for another important form.

Since $\mathrm{d}\theta^i$ is a 2-form, we may expand it in the basis $\theta^i$ itself:$$\mathrm{d}\theta^i=-\frac{1}{2}C^i{}_{jk}\theta^j\wedge\theta^k$$But the first structure equation gives$$\mathrm{d}\theta^i=-\omega^i{}_j\wedge\theta^j$$where $\omega^i{}_j=\Gamma^i{}_{kj}\theta^k$. Some algebra gives $C^i{}_{jk}=\Gamma^i{}_{jk}-\Gamma^i{}_{kj}$. Using $\nabla_XY-\nabla_YX=[X,Y]$ and the defintion of the Christoffel symbols as $\nabla_{e_i}e_j=\Gamma^k{}_{ij}e_k$ establishes the desired relation.