I want to show that any matrix $A \in SL(n, \mathbb{R})$, $n \geq 2$, (ie, $\det A =1$) can be written as $A= \exp s \ \exp k$, where $s$ is symmetric, and $k$ skew-symmetric, in the Lie algebra of the Lie group $$\mathfrak{sl}(n, \mathbb{R}) = \{ b \in \mathcal{M}_n (\mathbb{R}) : \text{tr} \ b =0 \} $$ and $$\exp: \mathfrak{sl}(n, \mathbb{R}) \to SL(n, \mathbb{R})$$ is the standard exponential of matrices.
So far I showed that $(\exp s)^t = \exp s$ and $(\exp k)^t = -\exp k$ and tried to use these relations but I couldn't really get anything.