This question is a reference request (or a detailed answer if anyone is willing to write one). I want to read a rigorous treatment of complex differential forms for Riemann surfaces (or maybe for more general complex manifolds), but I'm very picky about how I would like these things to be explained.
Let $M$ be a real smooth manifold. For me, a smooth differential $k$-form is a smooth section of the $k$th exterior cotangent bundle $\bigwedge^k T^{\ast}(M)$ of $M$, a smooth manifold whose underlying set is the disjoint union
$$\bigcup\limits_{m \in M} \bigwedge^k T^{\ast}_m(M)$$ over the points $m \in M$ of the $k$th exterior power of the contangent space of $M$ at $m$. A chart $(U,\varphi) \subset \mathbb R^n$ defines a canonical basis $dx_{i_1} \wedge \cdots \wedge dx_{i_k}, 1 \leq i_1 < \cdots < i_k \leq n$ of $\bigwedge^k T_m^{\ast}(M)$ for each $m \in \varphi(U)$, where $dx_1, ... , dx_n$ is the usual basis of $T_p^{\ast}(\mathbb R^n)$ for $p \in \mathbb R^n$.
Now let $M$ be a Riemann surface (or maybe more generally a complex manifold). Then $M$ is also naturally a real smooth manifold, and we have the space $\Omega^k(M)$ of smooth differential $k$-forms of $M$.
Is there a treatment of complex holomorphic/meromorphic forms on $M$ which explains them in terms of smooth differential forms on $M$?