Gaussian - symmetric case:
consider a statistically independent sample of N observations from a normal distribution.The mean (loc) of the population is the results on which you would like to make a prediction. You can report the value of the sample mean and as an error on it the standard error (SD/$\sqrt{N}$), i.e. $mean \pm SD/\sqrt{N}$.
Asymmetric case:
consider the same situation as before but with an exponential distribution. It sounds to me wrong to report an estimate of the error of the result, i.e. the mean, which is a symmetric $ \pm $, as in the case before.
Does it make sense to report $ \pm \,SD/\sqrt{N}$ in the case of a mean of an asymmetric distribution. If not, what is a choice that make sense as general as possible?