Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Can $f$ always be represented as $$ f(x)=\exp(F(x))x, $$ where $F:\mathbb{R}^n\to Mat_{n\times n}$?
Intuition/Direction Somehow it seems that $F$ should be related to the Jacobian if this is true but I'm not sure..