I know that $R^a_{bcd}=\partial_c\Gamma^a_{db}-\partial_d\Gamma^a_{cb}+\Gamma^a_{cf}\Gamma^f_{db}-\Gamma^a_{df}\Gamma^f_{cb}$. But I found that for $R_{abcd}=g_{ae}R^e_{bcd}$, the following formula holds.
\begin{align} R_{abcd}=\frac12\left(g_{ad,bc}-g_{bd,ac}-g_{ac,bd}+g_{bc,ad}\right)-g_{ef}\left(\Gamma_{ac}^e\Gamma_{bd}^f-\Gamma_{ad}^e\Gamma_{bc}^f\right) \end{align}
I think the Levi-Civita connection is assumed for this formula. How can I derive this formula for all indices lowered from $R^a_{bcd}$ above? Could anyone please help me?