I have been struggling to understand how picking an orthonormal frame for the tangent space of a Riemann surface with local coordinates ${x_1,x_2}$ simplifies the matrix of one forms associated to its Levi-Civita connection. In particular,
$$ \nabla X= \sum_h \left(dX^h + \omega_j^i X^j\right)\otimes\frac{\partial}{\partial x_h}$$
where, $\omega^i_j= \Gamma_{ik}^jdx_k$. Now, supposedly, if I pick an orthonormal frame the matrix of one forms $(\omega)_{ij} = \omega^i_j$ will become $0$ in the diagonal and antisymmetric. However, if I pick an orthonormal frame, then the metric is the identity matrix and all my Christoffel symbols (and hence the one forms $\omega^i_j$) vanish. Can some explain this to me? What am I thinking wrong?