Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?

1. $M$ intersects all the elements of ${\cal L}$, but
2. for all $m\in M$, the set $M\setminus\{m\}$ no longer intersects all the elements of $\cal L$.

In fact, I do not even know the answer for $n=2$.