  We apply the method of skew-orthogonal polynomials (SOP) in the complex plane
to asymmetric random matrices with real elements, belonging to two different
classes. Explicit integral representations valid for arbitrary weight functions
are derived for the SOP and for their Cauchy transforms, given as expectation
values of traces and determinants or their inverses, respectively. Our proof
uses the fact that the joint probability distribution function for all
combinations of real eigenvalues and complex conjugate eigenvalue pairs can be
written as a product. Examples for the SOP are given in terms of Laguerre
polynomials for the chiral ensemble (also called the non-Hermitian real
Wishart-Laguerre ensemble), both without and with the insertion of
characteristic polynomials. Such characteristic polynomials play the role of
mass terms in applications to complex Dirac spectra in field theory. In
addition, for the elliptic real Ginibre ensemble we recover the SOP of
Forrester and Nagao in terms of Hermite polynomials.
