All possible flat conformal metrics of dimension greater than 2

Question

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible one-to-one conformally flat transformations $\tilde{g}_{mn}=e^{2\varphi}g_{mn}$, where both metrics are flat. Flat means the Riemann tensor $R_{ijkl}$ is zero. $$ \tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g {~\wedge\!\!\!\!\!\!\bigcirc~} \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g \right)\right]_{ijkl} \right) $$