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We consider the group of orthogonal $(n+l)\times (n+l)$ matrices $O(n+l)$ equipped with the probability distribution given by the normalized Haar measure. We are interested in the probability that such random orthogonal matrices have no real eigenvalue after erasing the first $l$ columns and rows. This probability is given in terms of a Fredholm determinant consisting of a weighted Hankel matrix related to the Hilbert matrix. For $l=1$ we perform an asymptotic analysis of this Fredholm determinant and find that the probability in question behaves asymptotically as $n^{-3/8}$ when $n\to\infty$. We also comment on the asymptotic behaviour of a bigger class of Fredholm determinants related to Hankel matrices with jumps in the symbol and also on the relation of our model to roots of Kac polynomials. This is joint work with Mihail Poplavskyi and partially with Emilio Fedele.