= Solution
Dividing the coordinate tangents by their <scale factors of orthogonal coordinates> gives an <orthonormal basis>
$$
\mathbf e_\rho=(\cos\phi,\sin\phi,0),\quad
\mathbf e_\phi=(-\sin\phi,\cos\phi,0),\quad
\mathbf e_z=(0,0,1).
$$
Evaluate the Cartesian <cross products> componentwise:
$$
\mathbf e_\rho\times\mathbf e_\phi=(0,0,\cos^2\phi+\sin^2\phi)=\mathbf e_z,
$$
$$
\mathbf e_\phi\times\mathbf e_z=(\cos\phi,\sin\phi,0)=\mathbf e_\rho,
\qquad
\mathbf e_z\times\mathbf e_\rho=(-\sin\phi,\cos\phi,0)=\mathbf e_\phi.
$$
Consequently
$$
\boxed{\mathbf e_\rho\times\mathbf e_\phi=\mathbf e_z,\quad
\mathbf e_\phi\times\mathbf e_z=\mathbf e_\rho,\quad
\mathbf e_z\times\mathbf e_\rho=\mathbf e_\phi.}
$$
The ordered cylindrical <basis> is right-handed: its <scalar triple product> $\mathbf e_\rho\cdot(\mathbf e_\phi\times\mathbf e_z)$ equals $+1$.
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