Erd\H{o}s--Ko--Rado and Hilton--Milner theorems for two-forms

In this short note we show that both generalizations of celebrated Erd\H{o}s–Ko–Rado theorem and Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from the folklore puzzle about possible arrangements of an intersecting family of lines.