= Solution
Let $Q$ be the real <orthogonal matrix>. If $Qv=\lambda v$ for a nonzero complex <eigenvector>, then preservation of the complex Euclidean <norm> gives
$$
\|v\|^2=\|Qv\|^2=|\lambda|^2\|v\|^2,
$$
so $|\lambda|=1$. Its <characteristic polynomial> has real coefficients, hence nonreal <eigenvalues> occur in conjugate pairs. Such a pair is $e^{i\alpha},e^{-i\alpha}$, and the remaining real <eigenvalue> is $+1$ or $-1$.
If all three <eigenvalues> are real, each is $\pm1$; two have the same sign and can be paired as $e^{\pm i\alpha}$ with $\alpha=0$ or $\pi$. Thus in every case the spectrum has the form
$$
\boxed{\{e^{i\alpha},e^{-i\alpha},\varepsilon\},\quad\alpha\in\mathbb R,\quad\varepsilon=\pm1.}
$$
Their product shows that $\varepsilon=\det Q$.
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