general relativity - Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$

By employing the Cartan structural equations, one can 'read off' the components of the Ricci tensor and Riemann curvature tensor in the orthonormal basis. My question is: how do I take my curvature tensors in the orthonormal basis back to the coordinate basis after?