= Solution
The <special orthogonal group> is
$$
SO(n)=\{A\in M_n(\mathbb R):A^TA=I,\ \det A=1\}.
$$
If
$$
A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SO(2),
$$
its first column is a unit <vector>, say $(\cos\theta,\sin\theta)^T$. The second column is the unique unit <vector> perpendicular to it that gives positive <determinant>, namely $(-\sin\theta,\cos\theta)^T$. Thus
$$
A=\begin{pmatrix}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta
\end{pmatrix},
$$
so $A$ is rotation through $\theta$ about the origin.
For $A\in SO(3)$, its real characteristic <polynomial> of odd degree has a real <eigenvalue>. Every <eigenvalue> of an orthogonal <matrix> has <modulus> one, so every real <eigenvalue> is $\pm1$. Nonreal <eigenvalues> occur in conjugate pairs whose product is one, while $\det A=1$; if all <eigenvalues> are real, their product likewise forces at least one to be $+1$. Hence $A$ fixes a nonzero <vector> $u$. The plane $u^\perp$ is $A$-invariant, and the restriction to it is an orientation-preserving orthogonal map. By the $SO(2)$ result it is a planar rotation. Therefore $A$ is a rotation about the axis $\mathbb Ru$.
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