For $SL_2$ a cocycle is given by
$$
\sigma(g,h)=\left( \frac{x(gh)}{x(g)} , \frac{x(gh)}{x(h)} \right)
$$
where for $g=\left(\begin{array}{ll} a & b \\ c & d\end{array}\right)\in SL_2(\mathbb{C}((t)))$, we define $x(g)=c$ unless $c=0$ in which case $x(g)=d$. $(\cdot,\cdot)$ is the tame symbol.

I'll see if I can come up with a good reference. I've seen this stuff over local fields, where this is attributed to Kubota, and Kazhdan-Patterson's paper on Metaplectic Forms has this formula in it (actually for $GL_2$). I would be suprised if there was a usable formula for higher rank groups.