Suppose l1 and l2 are skew lines in the space E3. Is it true that lines PQ, where P ∈ l1, Q ∈ l2, sweep the whole space?

Question

Suppose l1 and l2 are skew lines in the space E3. Is it true that lines PQ, where P ∈ l1, Q ∈ l2, sweep the whole space?

My idea that it is not true due to the fact that we can provide counterexample: can take parallellines in 3d space and then the line PQ, where P ∈ l1, Q ∈ l2, can not sweep the whole space. Is that true or am I missing something?