Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be radial. Then can $f$ necessarily be represented as $$ A(\|x\|)x = f(x), $$ for some function $z\mapsto A(z)$ from $\mathbb{R}$ into $SO(n)$?

Here, by *radial* I mean that there exists some $f_0\in C^1(\mathbb{R},\mathbb{R}^{n})$ such that $f(x)=f_0(\|x\|)$. So I'm curious if we can assume that $f_0$ is acting by matrix multiplication...