  We study quantum bipartite systems in a random pure state, where von Neumann
entropy is considered as a measure of the entanglement. Expressions of the
first and second exact cumulants of von Neumann entropy, relevant respectively
to the average and fluctuation behavior, are known in the literature. The focus
of this paper is on its skewness that specifies the degree of asymmetry of the
distribution. Computing the skewness requires additionally the third cumulant,
an exact formula of which is the main result of this work. In proving the main
result, we obtain as a byproduct various summation identities involving
polygamma and related functions. The derived third cumulant also leads to an
improved approximation to the distribution of von Neumann entropy.
