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We apply the method of skew-orthogonal polynomials (SOP) in the complex planeto asymmetric random matrices with real elements, belonging to two differentclasses. Explicit integral representations valid for arbitrary weight functionsare derived for the SOP and for their Cauchy transforms, given as expectationvalues of traces and determinants or their inverses, respectively. Our proofuses the fact that the joint probability distribution function for allcombinations of real eigenvalues and complex conjugate eigenvalue pairs can bewritten as a product. Examples for the SOP are given in terms of Laguerrepolynomials for the chiral ensemble (also called the non-Hermitian realWishart-Laguerre ensemble), both without and with the insertion ofcharacteristic polynomials. Such characteristic polynomials play the role ofmass terms in applications to complex Dirac spectra in field theory. Inaddition, for the elliptic real Ginibre ensemble we recover the SOP ofForrester and Nagao in terms of Hermite polynomials.

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices