Let $(M,g)$ be a Riemmanian manifold. If a connection $\nabla$ is defined in a way such that $\Gamma_{ij,k}$ is given for all $i,j$ and $k$, how do we find $\Gamma_{ij}^k$? $\nabla$ is not necessarily Levi Civita connection here so we can't use $\Gamma_{ij}^k = \frac{1}{2} g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}) $.
Thanks