{c}
{title2=$L^{ij}=A^iB^j-B^iA^j$}

A line in real <projective space> $\mathbb P^3$ is the span of two independent homogeneous vectors $A,B$. Its Plücker coordinates are the entries of the displayed rank-two <antisymmetric matrix>, determined up to a nonzero scalar. They satisfy $L^{01}L^{23}-L^{02}L^{13}+L^{03}L^{12}=0$. With a chosen volume orientation, its dual plane form is $L_{ij}=\tfrac12\varepsilon_{ijkl}L^{kl}$. This is projective duality using the volume form, not lowering indices with a Euclidean metric.


 Plücker coordinates (source code)