Standard error of the variance of Uniformly distributed data

I would like to derive the standard error of the variance for a sample taken from a Uniform distribution with bounds $$a$$ and $$b$$.

The standard error of the mean of that sample is straigtforward: $$SE_{mean} = \frac{1}{\sqrt{3 n}} \frac{b-a}{2}$$ where $$n$$ is the sample size.

If possible, I am also looking for the standard error of the standard deviation.

How can I proceed?