Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$

Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$

$SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, \det(A) = 1}$}, which is the special orthogonal group. And $\mathbb{R}P^3$ is the real projective space.

I am trying to find an atlas for $SO(3)$. In order to do that, I would like to find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$. I know how to show that they are diffeomorphic (e.g.), but I am not sure how to find an explicit diffeomorphism.

Let $B$ be a closed 3-ball of radius $\pi$. Define the equivalence relation on $\partial B$ by $x\sim y$ if $x=-y$. Then $\mathbb RP^3$ can be considered as the quotient space $B/\sim$. So any point $[x]\in\mathbb RP^3$ is an equivalence class of a point $x$ in the ball. If $|x|<\pi$, then the equivalence class is $[x]=\{x\}$. If $|x|=\pi$, then $[x]=\{x,-x\}$.

$SO(3)$ is the set of all $3\times 3$ rotation matrices. We can specify a rotation by a direction in $\mathbb R^3$ along with a magnitude in $[-\pi,\pi]$. Of course, the a rotation by $\pi$ is equivalent to a rotation by $-\pi$. Therefore, we can represent any element of $SO(3)$ besides the identity by the ordered pair $(\vec v,m)$ where $\vec v$ is the unit vector in the direction of the rotation, and $m$ is the magnitude (between $-\pi$ and $\pi$). We represent the identity by $I$. The explicit diffeomorphism is $$f(\vec v,m)=[m\vec v],$$ and $$f(I)=0.$$

Now you just need to show that this is in fact a well-defined diffeomorphism.