Christoffel Symbol $\Gamma_{ii}^{i}$ in normal coordinates

Question

Chossing an orthonormal basis of $T_pM$ for $p \in (M,g)$ a Riemannian manifold I came across (e.g. here equation (7)) the expansion formula for the Christoffel symbols $$ \Gamma^{a}_{bc}(y)= \frac{1}{3} (R_{cdb}^{a} + R_{bdc}^{a})y_d + ... $$ where $R_{cdb}^{a}$ are the coefficients of the curvature tensor in $p$ and $|y|$ small enough such that $\exp_p y $ is in a geodesic neighborhood of $p$. If now $a=b=c$ we have $$ (R_{cdb}^{a} + R_{bdc}^{a})y_d=0. $$ So the Christoffel symbols $\Gamma_{ii}^{i}(y)$ for $1\leq i \leq n$ vanish up to the second order terms. I was wondering, if there are antisymmetries of the Christoffel symbols that actually yield that $$ \Gamma_{ii}^{i}(y)=0. $$ Or are there counterexamples?