variance of the sample variance

Apologies if it is a trivial question, I googled but didn't find a direct answer. I'd like to check my guess

I am given a random sample from a normal distribution with variance $\displaystyle \sigma^2$. I need to find variance of the sample variance, ie $\displaystyle Var(S^2)$.

I use the fact that, for a random sample from a normal population with variance $\displaystyle \sigma^2$,

$\displaystyle \frac{(n-1)S^2}{\sigma^2}\sim{\chi_{n-1}^2$

I found in the references that in the chi-squared distribution of n-1 degrees of freedom, the variance is 2(n-1)*

Therefore

$\displaystyle Var(\frac{(n-1)S^2}{\sigma^2})=2(n-1)$

$\displaystyle \frac{(n-1)^2}{\sigma^4}Var(S^2)=2(n-1)$

then find $\displaystyle Var(S^2)=\frac{2{\sigma}^4}{n-1}$

Is that so?

*By the way, is there a link somewhere to a simple derivation of the Variance (and the Mean) formula for the chi-squared distribution? My textbook at hand just quotes it without proof.