Covariant derivative along higher order tensor

Question

I've recently started to review some Riemann calculus and I've got a question that I could not find anywhere else. So, the covariant derivative along a vector field is defined as: \begin{equation} \nabla_{\vec{v}}=v^{i}\nabla_{\vec{e}_{i}} \end{equation} So far, so good. My question is how can we define a covariant derivative along a rank-2 tensor, i.e., what is the definition of: \begin{equation} \nabla_{T}=T^{ij}\nabla_{\vec{e}_i\vec{e}_j} \quad,\quad \nabla_{\vec{e}_i\vec{e}_j}=? \end{equation}