To deduce the Cartan structural equations in a coordinated frame we are going to use the definition of the Christoffel symbols (connection coefficients) and where we always are going to use the Einstein sum convention: $$\nabla_{\partial_ i}\partial_j={\Gamma^s}_{ij}\partial_s$$ and the curvature tensor $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ where $X,Y,Z$ are any three vector fields in a riemannian manifold $\cal{M}$ with the Levi-Civita connection $\nabla$ .

First, we define through the relation $\nabla_X\partial_i={\omega^s}_i(X)\partial_s$ a set of scalar function ${\omega^s}_i$ which are easily to see that they actually are 1-forms. We observe that ${\omega^s}_i(\partial_j)={\Gamma^s}_{ij}$ .

They satisfy skew-symmetry rule: $\omega_{si}=-\omega_{is}$ , which arises from the covariant constancy of the metric tensor $g_{kl}$ i.e. \begin{eqnarray*} 0&=&\nabla_{X}g_{kl}\\ &=&\nabla_{X}\langle\partial_k,\partial_l\rangle\\ &=&\langle\nabla_{X}\partial_k,\partial_l\rangle+\langle\partial_k,\nabla_{X}\partial_l\rangle\\ &=&\langle{\omega^s}_k(X)\partial_s,\partial_l\rangle+\langle\partial_k,{\omega^s}_l(X)\partial_s\rangle\\ &=&{\omega^s}_k(X)g_{sl}+{\omega^s}_l(X)g_{ks}\\ 0&=&\omega_{lk}(X)+\omega_{kl}(X) \end{eqnarray*}that last equation is valid for each vector field $X$ , then $\omega_{lk}=-\omega_{kl}$ .

Next we define through the relation $$R(X,Y)\partial_i={\Omega^s}_i(X,Y)\partial_s$$ the scalars ${\Omega^s}_i(X,Y)$ which are the so called connection 2-forms. That they are really 2-forms is an easy caligraphic exercise.

Now by the use of the Riemann curvature tensor above we see \begin{eqnarray*} R(X,Y)\partial_i&=&\nabla_X\nabla_Y\partial_i-\nabla_Y\nabla_X\partial_i-\nabla_{[X,Y]}\partial_i\\ &=&\nabla_X({\omega^s}_i(Y)\partial_s)-\nabla_Y({\omega^s}_i(X)\partial_s)-{\omega^s}_i[X,Y]\partial_s\\ &=&X({\omega^s}_i(Y))\partial_s+{\omega^s}_i(Y)\nabla_X\partial_s- Y({\omega^s}_i(X)\partial_s-{\omega^s}_i(X)\nabla_Y\partial_s-{\omega^s}_i[X,Y]\partial_s\\ &=&X({\omega^s}_i(Y))\partial_s+{\omega^s}_i(Y){\omega^t}_s(X)\partial_t- Y({\omega^s}_i(X)\partial_s-{\omega^s}_i(X){\omega^t}_s(Y)\partial_t-{\omega^s}_i[X,Y]\partial_s\\ &=&[X({\omega^s}_i(Y))+{\omega^t}_i(Y){\omega^s}_t(X)-Y({\omega^s}_i(X))-{\omega^t}_i(X){\omega^s}_t(Y)-{\omega^s}_i[X,Y]]\partial_s\\ {\Omega^s}_i(X,Y)\partial_s &=& [X({\omega^s}_i(Y))-Y({\omega^s}_i(X))-{\omega^s}_i[X,Y]+{\omega^s}_t(X){\omega^t}_i(Y)-{\omega^s}_t(Y){\omega^t}_i(X)]\partial_s\\ \end{eqnarray*}In this last relation we recognize -in the first three terms- the exterior derivative of ${\omega^s}_i$ evaluated at $(X,Y)$ i.e. $$d{\omega^s}_i(X,Y)=X({\omega^s}_i(Y))-Y({\omega^s}_i(X))-{\omega^s}_i[X,Y]$$ and in the last two terms its wedge product $${\omega^s}_t\wedge{\omega^t}_i(X,Y)={\omega^s}_t(X){\omega^t}_i(Y)-{\omega^s}_t(Y){\omega^t}_i(X)$$ all these for any two fields $X,Y$ . Hence $${\Omega^s}_i=d{\omega^s}_i+{\omega^s}_t\wedge{\omega^t}_i$$ which is called the second Cartan structural equation for the coordinated frame field $\partial_i$ .

More interesting things happen in an an-holonomic basis.