Certain polynomials of a skew-symmetric matrix are considered. These polynomials can be expressed in terms of the zonal polynomials on the Hermitian matrices, and they are used to obtain a series expansion for the density of the non-null distribution of the maximal invariant corresponding to the problem of testing for reality of the covariance matrix of a complex multivariate normal distribution.
Aksel Bertelsen. "On Non-Null Distributions Connected with Testing Reality of a Complex Normal Distribution." Ann. Statist. 17 (2) 929 - 936, June, 1989. https://doi.org/10.1214/aos/1176347152