The Central Limit Theorem's rate of convergence is $O\left(1/\sqrt{n}\right)$. There are simple distributions for which this rate convergence cannot be improved, for example if the distributions are i.i.d. Bernoulli trials. There are some special cases, however, where the convergence rate can be improved, for example if the distributions are i.i.d. uniform distributions (in which case the rate is $O(1/n)$), and an extreme case is if all distributions are normal (in which case the CLT is exact).
My question is: is there a simple "test" one could apply to a probability distribution to estimate the rate of convergence of $\frac{\sum_{i=1}^nX_i}{n}$ to a normal distributions, when $\left(X_i\right)_i$ is a family of i.i.d. random variables from this distribution? Given the examples above, it should probably be about close the $X_i$ are to Gaussian distribution, or maybe how much of them is centered in the tails vs. the center?