Enumerative Geometry - General equation of the transversal line intersecting 3 skew lines in 3D

Sketch: Call the respective lines $\ell_1$, $\ell_2$, and $\ell_3$. For each point $P\in\ell_1$, consider the plane $\Pi_P$ determined by $P$ and $\ell_2$. The line $\ell_3$ will (generically) meet $\Pi_P$ in a unique point $Q_P$. [You will, of course, want to use different variables as parameters for $\ell_2$ and $\ell_3$.] The locus of the lines $\overleftrightarrow{PQ_P}$ (as $P$ varies over $\ell_1$) will be your ruled surface.

Note that you can get everything explicitly parametrically here, using $t$ to parametrize $\ell_1$—and hence the pencil of lines $\overleftrightarrow {PQ_P}$—and $s$ as the parameter along those lines.

Comment: If you want an example where things work out rather nicely, take instead the lines \begin{align*} \ell_1: \quad & x=t, \ y=z=0 \\ \ell_2: \quad & x=t+1,\ y=1,\ z=t \\ \ell_3: \quad & x=t+1,\ y=2,\ z=2t+2 \end{align*}