2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/118022We find a local $(d+1) \times (d+1)$ Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree $d$. Our Riemann-Hilbert problem is similar to a local $d \times d$ Riemann-Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann-Hilbert problems, and brings us closer to finding asymptotics of the skew-orthogonal polynomials.9 pages, v. 2 fixed some typos, v. 3 revised proofComplex VariablesMathematical Physics15A52 (33C45; 82B31)text