I think you're looking for Liouville's theorem. 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 higher-dimensional analogue of a Mobius transformation.
By the way, observe that there are no assumptions on the topology of the $V_i$ -- they don't have to be simply-connected, etc.