  We 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.
