Please allow me to address the set cover problem approach in a new answer, so the question won't introduce more confusions. I would update this answer as I venture through this path.
And first, I have to explain myself a bit more. I've deserted maths for a decade more (not intentionally though). So my maths ability drops to every-day-maths capable. I'm picking stuffs up as I progress.
It seems I have made a wrong choice regarding $L_{j}$. As my goal is to select crew that can fulfill all jobs, $L_{j}$ won't help here. Instead list of doable jobs per worker should be the one to get.
$U \leftarrow$ all jobs
$W \leftarrow$ all workers
$L_w \leftarrow$ jobs can be done by worker $w$ (where $w \in W$ and $L_w \in U$)
Grep the set cover ILP from wiki: $$ \mathrm{minimize} \sum_{j \in L_w} x_j $$
for all $e \in U$ $$ \sum_{j_{:e} \in L_w} x_j \ge 1 $$
for all $j \in L_w$ $$ x_j \in \{0,1\} $$
now we have the combination of workers (first part solved), correct?
Below stuffs were turned out unused.
$Z \leftarrow$ slots
$n_u = |U|$
$u_j \in U$ (where $1 \le j \le n_u$)
$n_w = |W|$
$w_i \in W$ (where $1 \le i \le n_w$)
$S \leftarrow$ all skills & abilities
$s_j \leftarrow$ job $j$ skills, $s_j \in S$ (where $1 \le j \le n_u$)
$S' \in S$ (all skills used in jobs) where: $$ S' = \bigcup_{j=1}^{n_u} s_j $$
$s_i \leftarrow$ worker $i$ skills, $s_i \in S' \in S$ (where $1 \le i \le n_w$)
$L_j \leftarrow$ suitable workers to job $j$ (where $j \in U$)
$v_i \leftarrow$ worker $i$ score