In terms of local coordinates we can make a covariant derivative exterior-derivative-like (this is actually Levi-Civita connection) \begin{equation} \Gamma_{i,j}^k = \Gamma_{j,i}^k \implies D(dx_k) = \sum_{ij}{\Gamma_{i,j}^k dx_i \otimes dx_j} = 0 = d(dx_k) = \sum_j{\frac{\partial{1}}{\partial{x_j}} dx_j \wedge dx_k} \end{equation} Could I then say, based on this observation, that we can view exterior derivative as a special case of covariant derivative?
By the way note the difference between ''$\otimes$'' for covariant derivative and ''$\wedge$'' for exterior derivative. One is defined on tensor algebra and the other is on exterior algebra. And we know that exterior algebra comes out of tensor algebra by quotienting out the two-sided ideal generated by $\{a \otimes b + b \otimes a\}$. Will this domain perspective also back the claim that exterior derivative can be considered as a special case of covariant derivative?