How can we compute the skewness coefficient of the sample variance given by $E(\frac{[S^2-E(S^2)]^3}{[Var(S^2)]^{3/2}})$ in terms of the moments of X?

Question

I have been trying to compute the skewness coefficient of the sample variance given by $E(\frac{[S^2-E(S^2)]^3}{[Var(S^2)]^{3/2}})$. I have used $E(S^2)=\sigma^2$ and I have ended up with $\frac{E(S^6)-3\sigma^2Var(S^2)-\sigma^6}{[Var(S^2)]^{3/2}}$ where $Var(S^2)= \frac{1}{n} [E(X-\mu)^4-(\frac{n-3}{n-1})\sigma^4]$. How do we proceed? How can $E(S^6)$ be computed? Can my result be simplified so as to be expressed in terms of just $\mu$, $\sigma$, n and $E(X-\mu)^4$? I would like the result to be more general and not depending on the sample variance which is different each time. So I thought of using $S^2=\frac{\Sigma(x_i-\bar{x})^2}{n-1}$ but then I got many terms one of which was $E[(\Sigma x_i^2)^6]$. How can this be expressed in terms of the moments of X? Thank you in advance for your help.