As is commonly known, the connection 1-forms of a Riemannian manifold are skew-symmetric: $\omega^i_j=-\omega^j_i$. Until now, I have not actually thought to hard on this, but I think I've hit a snag.
Let's consider the metric $e^{-(x^2+y^2)}(dx^2+dy^2)$ on $\mathbb{R}^2$. Then, by computation, the Christoffel symbols of the Levi-Civita connection are $\Gamma^1_{11}=\Gamma^2_{12}=-\Gamma^1_{22}=x$ and $-\Gamma^2_{11}=\Gamma^1_{12}=\Gamma^2_{22}=y$. Organizing these into a matrix 1-form gives us $$\begin{bmatrix}xdx+ydy & ydx-xdy \\ xdy-ydx & xdx+ydy\end{bmatrix},$$ which is not skew symmetric on the diagonal (i.e. nonzero on the diagonal).
What have I done wrong here? Have I simply misinterpreted something? I've double checked the calculations, so I'm fairly certain it is not a computation problem.
To be clear, I know how to prove that they are skew-symmetric using the fact that their inner products will be constant. I simply want to know what I did wrong in this instance.