Hyperbolic Metric on a Riemann Surface

Question

From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the punctures correspond to fixed points of the parabolic elements of the associated Fuchsian group. The question is that: what is the explicit local expression of this unique hyperbolic metric for such a surface around a puncture, associated with the fixed point $x$ of a parabolic element? A good reference is highly appreciated.