If $\omega_i$ are dual basis one forms corresponding to an orthonormal tetrad basis $e_i$, and given that the commutation coefficients $C_{ij}^k$ are defined by
\begin{equation}[e_i,e_j]=C_{ij}^k e_k\end{equation}
how do you prove the following equation
\begin{equation}\mathrm{d}\omega^a=-\frac{1}{2}C_{bc}^a\omega_b\wedge\omega_c\end{equation}
as given in equation 5.6, page 99 of "Relativity Demystified" by David McMahon, published by McGraw Hill, 2006, and where $\mathrm{d}$ is the exterior derivative operator?