Let $L,M,N$ be skew lines in $\mathbb{P}^3$ am asked to find the lines in $\mathbb{P}^3$ which meet all three these lines.
I think that, being lines in $\mathbb{P}^3$ the projectivization of two-dimensional spaces, we can write, up to projective equivalence
$L=[a_0:a_1:0:0]$
$M=[0:0:b_1:b_2]$
$N=[c_0:c_1:c_0:c_1]$
where the indexes $a_i,b_i,c_i$ are never simultaneously $0$.
Now I am lead to think that a projective line meeting all the three should be of the form $[d_0:d_1:d_2:d_3]$ with all $d_i$ different from $0$.
But this does not seem a line either!
How can I solve this problem ? What can we say in general about skew $k$-planes in $\mathbb{P}^{2k-1}$?