Almost skew-symmetric matrices are real matrices whose
symmetric parts have rank one. Using the notion of the numerical range, we
obtain eigenvalue inequalities and a localization of the spectrum of an
almost skew-symmetric matrix. We show that almost skew-symmetry is
invariant under principal pivot transformation and inversion, and that
the symmetric parts of Schur complements in almost skew-symmetric matrices
have rank at most one. We also use affine combinations of
and
to
gain further insight into eigenvalue location and the numerical range of an
almost skew-symmetric matrix.