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 . I need to find variance of the sample variance, ie .

I use the fact that, for a random sample from a normal population with variance ,

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

Therefore

then find

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.