Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of $\pi_1(\Sigma_g)$ can be thought of as a biholomorphic map $~f: \mathbb{H} \rightarrow \mathbb{H}$ and $\mathbb{H}$ is the upper half plane with the Hyperbolic metric.
Moreover, for a fixed $\Sigma_g$ there will be different families of maps representing $\pi_1(\Sigma_g)$, which will induce distinct complex structures on $\Sigma_g$.
I am looking for a reference where these maps are explicitly written down, for $g \geq 2 $.
$\textbf{EDIT:}$ The question as stated initially was wrong as has been pointed out. A genus $g$ Riemann surface is the quotient of the Hyperbolic space and not the complex plane.