In the case of distribution with a large population size $N \ge 10000$, which follows an unusual distribution with stand deviation $\sigma$. By the central limit theorem, repeat sampling with a large sampling size $n \geq 30$, the sample mean $\mu_{\bar{x}}$ would follow the sample distribution $~\mathcal{N}(\mu_{\bar{x}}, \sigma_{\bar{x}})$ where $\sigma_{\bar{x}} = \sigma/\sqrt{n}$ is the standard deviation of the sampling distribution.
However, for a small sampling size $n\leq 10$, is there a proper way to calculate the standard deviation of the sampling distribution $\sigma_{\bar{x}}$ by directly calculating from the given population instead of sampling? Since the sampling distribution does not exactly follow a gaussian for small $n$.