I am preparing for an exam, and would like to have a rigorous definition of the following:

**Orientation-preserving isometry of $R^n$**

I know that it is something like the following (feel free to correct my wording):

When the homomorphism  $\pi:M_n \rightarrow O_n$ is applied to the unique representation [TEX] t_a \phi [\TEX] of an isometry f, and [TEX] \pi(f)=\phi [\TEX], define [TEX] \sigma:M_n \rightarrow \pm 1 [\TEX]. This map that sends an isometry of [TEX] R^n [\TEX] to $1$ is **orientation-preserving**.