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


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

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

where the product is taken over $j< k$ and 

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

(Sorry for the typesetting clumsiness. There seems to be  problem with MathJax.) 

Now use this formula for 

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

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


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