Unbiased estimator involving chi square distribution

Let X1,...,Xn denote a random sample from a normal distribution with mean zero and variance $\displaystyle \theta$. Show that $\displaystyle \Sigma \frac{X^2_i}{n}$ is an unbaised estimator of $\displaystyle \theta$ and it's variance is $\displaystyle \frac{2\theta^2}{n}$

I know that $\displaystyle \frac{X^2_i}{\theta}$ is a chi square distribution with n degrees of freedom, and hence, it's expectation is just n.

I'm just not sure how to use this fact.

Any help would be appreciated.