For completeness, econometricians also like the Kiefer and Salmon test from their 1983 paper in Economics Letters -- it sums 'normalized' expressions of skewness and kurtosis which is then chi-square distributed. I have an old C++ version I wrote during grad school I could translate into R.
Edit: And here is recent paper by Bierens (re-)deriving Jarque-Bera and Kiefer-Salmon.