We study the asymptotic behavior of Mardia's measure of (sample) multivariate skewness. In the special case of an elliptically symmetric distribution, the limit law is a weighted sum of two independent $\chi^2$-variates. A normal limit distribution arises if the population distribution has positive skewness. These results explain some curiosities in the power performance of a commonly proposed test for multivariate normality based on multivariate skewness.
"Limit Distributions for Mardia's Measure of Multivariate Skewness." Ann. Statist. 20 (4) 1889 - 1902, December, 1992. https://doi.org/10.1214/aos/1176348894