Linearization of a Isometry and a bounded function.

Question

I am here to ask for the linearization of the following function, let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f(x)=\Psi(x)+\rho(x)$ where $\Psi$ is a isometry in $\mathbb{R}^n$ and $\rho$ is a bounded function and $C^1.$ Searching for some information I found the Hartman-Grobman theorem but I think in this case (or for example in $\mathbb{R}^2$) find a linearization for $f$ it can be simpler.