Such maps are called conformal. A theorem of Liouville says that if $n\geq 3$, the only conformal maps (defined in some region in $R^n$) are Mobius. A Mobius map is a composition of inversions in spheres. For example $x\mapsto x/|x^2|$ is the inversion in the unit sphere. Inversions in all spheres generate the Mobius group.
This is not so in dimension $2$. Any complex analytic function whose derivative is not equal to zero defines a conformal map in dimension 2.