  Let $A$ be a real skew-symmetric Gaussian random matrix whose upper
triangular elements are independently distributed according to the standard
normal distribution. We provide the distribution of the largest singular value
$\sigma_1$ of $A$. Moreover, by acknowledging the fact that the largest
singular value can be regarded as the maximum of a Gaussian field, we deduce
the distribution of the standardized largest singular value
$\sigma_1/\sqrt{\mathrm{tr}(A'A)/2}$. These distributional results are utilized
in Scheff\'{e}'s paired comparisons model. We propose tests for the hypothesis
of subtractivity based on the largest singular value of the skew-symmetric
residual matrix. Professional baseball league data are analyzed as an
illustrative example.
