Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are global or not. How can I know if a parametrization obtained by exponentiating Lie algebra generators is a global parametrization?

As a second part of this question, SL(2,R) can be easily parametrized with three variables $t,\phi, \chi$ as $$\left[\begin{array}{cc}\cosh\chi\cos t+\sinh\chi\sin\phi&\sinh\chi\cos\phi+\cosh\chi\sin t\\\sinh\chi\cos\phi-\cosh\chi\sin t&\cosh\chi\cos t-\sinh\chi\sin\phi\end{array}\right]$$

Are there similar parametrizations for SL(3,R), that have reasonably simple expressions for the elements of the matrix? If I try to exponentiate the Lie algebra elements the resulting matrix elements are a mess.