Christoffel symbol matrix formulation

Question

Let there exist an arbitrary (2,0)-tensor $T^{ij}$ such that the covariant derivative is defined as $$\nabla_aT^{ij}=\partial_aT^{ij}+\Gamma^i_{ak}T^{kj}+\Gamma^j_{ak}T^{ik}. $$ To define the Christoffel there must exist some metric $g_{ij}.$ The Christoffel symbol defined in terms of the metric connection follows $$\Gamma^i_{ak}=\frac{1}{2}g^{im}(\partial_kg_{ma}+\partial_ag_{mk}-\partial_mg_{ak}).$$ Since $k$ is the dummy index in the covariant derivative it will take values $$\Gamma^i_{at},\Gamma^i_{ar},\Gamma^i_{a \theta},\Gamma^i_{a \phi}$$. I know how to represent the Christoffel symbols in matrix form to find out all the different various permutations of indices, like $$\Gamma^t_{ai}=\begin{pmatrix}\Gamma^t_{00}&\Gamma^t_{01}&\Gamma^t_{02}&\Gamma^t_{03}\\\Gamma^t_{10}&\Gamma^t_{11}&\Gamma^t_{12}&\Gamma^t_{13}\\\Gamma^t_{20}&\Gamma^t_{21}&\Gamma^t_{22}&\Gamma^t_{23}\\\Gamma^t_{30}&\Gamma^t_{31}&\Gamma^t_{32}&\Gamma^t_{33}&\end{pmatrix},$$ where the 1st covariant index $a$ represents the row and $i$ the column. If there exists a dummy index in one of the covariant indices like $$\Gamma^i_{ak}$$ where $k$ is the dummy indices and takes values $$\Gamma^i_{at},\Gamma^i_{ar},\Gamma^i_{a \theta}, \Gamma^i_{a \phi}.$$ How would I represent $\Gamma^i_{at}$ in matrix form where t is the index that is fixed and does not represent any row or column? Would the $i$ index be the row index then $a$ the column index or vice versa such that it follows as $$\Gamma^i_{at}=\begin{pmatrix}\Gamma^0_{0t}&\Gamma^0_{1t}&\Gamma^0_{2t}&\Gamma^0_{3t}\\\Gamma^1_{0t}&\Gamma^1_{1t}&\Gamma^1_{2t}&\Gamma^1_{3t}\\\Gamma^2_{0t}&\Gamma^2_{1t}&\Gamma^2_{2t}&\Gamma^2_{3t}\\\Gamma^3_{0t}&\Gamma^3_{1t}&\Gamma^3_{2t}&\Gamma^3_{3t}\end{pmatrix}?$$ Would that be the right idea or would I have to swap the top and bottom indices?