Let $P\in{\mathbb R}[X]$ be a monic polynomial with roots on the unit circle. For the problem below, we may assume wlog that the roots are simple and distinct from $\pm1$. It can be shown that there exists a matrix $M\in{\bf SO}_n({\mathbb R})$, whose characteristic polynomial is $P$ (an orthogonal companion matrix of $P$, in short OCM). See for instance Exercise 99 on my list http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf . Regretfully, this exercise uses the square root of Hermitian positive definite matrices, which cannot be computed in finitely many operations.