Riemann Curvature tensor for surfaces

Question

Let $M$ be a regular surface on $\mathbb{R}^3$. I am trying to express the Riemann's curvature tensor: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ respect $R(\vec x_i,\vec x_j)\vec x_k$. It is easy to show: $$R(X,Y)Z=\sum_{i,j,k=1}^n\Big(\nabla_{X^i\vec x_i}\nabla_{Y^j\vec x_j}Z^k\vec x_k-\nabla_{Y^j\vec x_j}\nabla_{X^i\vec x_i}Z^k\vec x_k\Big)-\nabla_{[X,Y]}Z$$ What I can do with: $$\nabla_{[X,Y]}Z\ ?$$ Many thanks!