= Solution
Orient the axis towards $n=(3/5,4/5,0)$ and measure positive angles by the <right-hand rule> about $n$. The <vectors> $n$, $m=(-4/5,3/5,0)$ and $e_3=(0,0,1)$ form a positively oriented <orthonormal basis>, since $n\times m=e_3$. Its <change-of-basis matrix> is
$$
Q=\begin{pmatrix}3/5&-4/5&0\\4/5&3/5&0\\0&0&1\end{pmatrix},\qquad Q^{-1}=Q^T.
$$
Relative to this <basis>, the <rotation> fixes the first coordinate and rotates the last two. Writing $c=\cos\alpha$ and $s=\sin\alpha$, its <matrix> in the <standard basis> is therefore
$$
\boxed{R=Q\begin{pmatrix}1&0&0\\0&c&-s\\0&s&c\end{pmatrix}Q^T
=\begin{pmatrix}(9+16c)/25&12(1-c)/25&4s/5\\12(1-c)/25&(16+9c)/25&-3s/5\\-4s/5&3s/5&c\end{pmatrix}.}
$$
In particular $Rn=n$, $Rm=cm+se_3$, and $Re_3=-sm+ce_3$, verifying the axis and the direction of <rotation>. Reversing the chosen orientation of the axis replaces $\alpha$ by $-\alpha$.
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