We have \begin{align*} \mathfrak{so}(n)&=\{B \in \mathfrak{gl}(n, \mathbb{R});\,\, B+B^{T}=0\}\\ SO(n)&=\{B \in M(n, \mathbb{R}); \,\,\det B=1 \quad \mbox{e} \quad B(B^{T})=I \}, \end{align*} where $\mathfrak{so}(n)$ is the Lie algebra of the Lie group $SO(n).$
My problem is
Show that $f:\mathfrak{so}(n)\rightarrow SO(n)$ given by$$A \rightarrow (I-A)(I+A)^{-1}$$ define a homeomorphism between an neighborhood of zero in $\mathfrak{so}(n)$ and an neighborhood of $Id_{n}$ in $SO(n)$.
I have no idea where to start this issue. I would like an answer. Thanks in advanced.