Plücker coordinates and grassmannians

Question

Let $X$ denote the set of lines in $\mathbb{P}_k^3$ and $\varphi:X\rightarrow\mathbb{P}_k^5$ be the morphism that sends, once we have chosen a reference, each $(a_0:\cdots:a_3)\vee(b_0:\cdots:b_3)$ to $$(\Delta_{01}:\Delta_{02}:\Delta_{03}:\Delta_{12}:\Delta_{13}:\Delta_{23}),$$ where each $\Delta_{ij}$ is given by the determinant $$ \Delta_{ij}=\left|\begin{array}{cc} a_i & a_j \\ b_i & b_j\end{array}\right| $$ Which is the relationship between the Plücker coordinates of a line and the intersections of such line with the reference tetrahedron? I am aware that in many textbooks the approach to this problem is exterior algebras. Is there an elementary, alternative way of seeing that relationship? On the other hand, would that relationship imply that $\varphi$ is injective? If so, how?