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In case anyone else comes across this: the model I ended up using is by León, Massé, and Rivest in the Journal of Multivariate Analysis (see here). They give a distribution on the space of skew-symmetric matrices that gives an arbitrarily concentrated distribution on the orthogonal group: that is, for dispersion parameter $\kappa=0$ the resulting distribution is uniform, and as $\kappa\rightarrow\infty$ the distribution approaches a constant distribution.

EDIT: The limit is a Gaussian distribution (analogous to the degrees of freedom of a t distribution approaching $\infty$), not constant.

Thanks again to those who commented/answered as these suggestions put me on the path towards the method I ultimately found.

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In case anyone else comes across this: the model I ended up using is by León, Massé, and Rivest in the Journal of Multivariate Analysis (see here). They give a distribution on the space of skew-symmetric matrices that gives an arbitrarily concentrated distribution on the orthogonal group: that is, for dispersion parameter $\kappa=0$ the resulting distribution is uniform, and as $\kappa\rightarrow\infty$ the distribution approaches a constant distribution.

Thanks again to those who commented/answered as these suggestions put me on the path towards the method I ultimately found.