I know that the set of all lines in $\mathbb{P}^3$ is given by $$p_{0,1}p_{2,3}-p_{0,2}p_{1,3}+p_{0,3}p_{1,2}=0 \ \ \ (*)$$ where $(p_{0,1}, p_{0,2}, p_{0,3}, \cdots, p_{2,3})$ is the Plucker coordinates in $\mathbb{P}^5$.
How do you get the Plucker coordinates of some subset of lines in $\mathbb{P}^3$? For example let's consider the set of lines through $P \in \mathbb{P}^3$.
I know that the set of lines through $P$ can be identified with $\mathbb{P}^2$, and according to the table A.1: Varieties of degree 1 in projective space here (http://robotics.technion.ac.il/people/alon/Appendix%20A.htm), I must find two linear equations in terms of $p_{ij}$ that every element in the set satisfies. We are given the (homogeneous) coordinates of $P$, say $(a, b, c, d)$, but I can't think of the way to get the two more linear equations.