  The Seidel matrix of a tournament on $n$ players is an $n\times n$
skew-symmetric matrix with entries in $\{0, 1, -1\}$ that encapsulates the
outcomes of the games in the given tournament. It is known that the determinant
of an $n\times n$ Seidel matrix is $0$ if $n$ is odd, and is an odd perfect
square if $n$ is even. This leads to the study of the set \[ \mathcal{D}(n)= \{
\sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. \] This paper
studies various questions about $\mathcal{D}(n)$. It is shown that
$\mathcal{D}(n)$ is a proper subset of $\mathcal{D}(n+2)$ for every positive
even integer, and every odd integer in the interval $[1, 1+n^2/2]$ is in
$\mathcal{D}(n)$ for $n$ even. The expected value and variance of $\det S$ over
the $n\times n$ Seidel matrices chosen uniformly at random is determined, and
upper bounds on $\max \mathcal{D}(n)$ are given, and related to the Hadamard
conjecture. Finally, it is shown that for infinitely many $n$, $\mathcal{D}(n)$
contains a gap (that is, there are odd integers $k<\ell <m$ such that $k, m \in
\mathcal{D}(n)$ but $\ell \notin \mathcal{D}(n)$) and several properties of the
characteristic polynomials of Seidel matrices are established.
