I think you're looking for <a href="http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)">Liouville's theorem</a>.  This theorem states that for $n >2$, if $V_1,V_2 \subset \mathbb{R}^n$ are open subsets and $f : V_1 \rightarrow V_2$ is a smooth conformal map, then $f$ is the restriction of a Mobius transformation.