Unbiased estimator for UNIF($-\theta, \theta$)

Question

I am searching for an unbiased estimator for $\theta$ in a UNIF($-\theta, \theta$) distribution, which looks like $\hat\theta = c(X_{n:n} - X_{1:n}$). The question is to search for the c that makes this estimator unbiased. I already found that $c = \cfrac{\theta}{E(X_{n:n} - X_{1:n})}$, because then $E(\hat\theta) = \theta$. By trial and error, I found that $E(X_{n:n} - X_{1:n}) = \cfrac{n+1}{n-1}*2\theta$ just by trying it for different values for n. But has anyone an idea how to proof this? I indeed get the right answer, but I don't now how to formally proof that $E(X_{n:n} - X_{1:n}) = \cfrac{n+1}{n-1}*2\theta$