There is  a     Weyl integration  formula that deals with this problem.  $\newcommand{\bR}{\mathbb{R}}$ Denote by  $G_n$   the space of symmetric $n\times n$ real matrices.  It states  that for any  $O(n)$-invariant  $h:G_n\to \bR$ we have


$$ (2\pi)^{-\frac{n(n+1)}{4}} \int_{G_n} h(A) e^{-\frac{{\rm tr} A^2}{4}} dA $$

$$
=\frac{1}{Z_n}\int_{\mathbb{R}^n} h(\lambda_1,\dotsc, \lambda_n) e^{-\frac{1}{4} \sum_{j=1}^n \lambda_j^2} \; {\prod_{j<k}} |\lambda_j-\lambda_k| \; d\lambda_1\cdots d\lambda_n$$

where 

$$Z_n =2^{\frac{n}{2}}n! \prod_{j=1}^n \Gamma(\frac{j}{2}).$$



Now, use this formula for 

$$ h(A)= f(g(A))\, e^{\frac{{\rm tr} A^2}{4}}. $$

For more details  see Appendix C of  [this paper][1] and the references therein.


  [1]: http://www.nd.edu/~lnicolae/RandCrVal.pdf