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Let's say I have $M$ samples of $N\times N$ real symmetric orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over $O(N)$?

This question is related to this previous question.

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# Statistics for Haar measure of random matrices?

Let's say I have $M$ samples of $N\times N$ real symmetric orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over $O(N)$?

This question is related to this previous question.