"Projecting" generally placed hyperplane, still generally placed?

Question

$\newcommand\Q{\mathbb Q} \renewcommand{\P}{\mathbb P}$Suppose I have $4$ hyperplanes (defined over $\Q$) that are in general (linear) position in $\P^n(\Q)$ with $n\geq 2$. I suspect that I can always find a two dimensional linear space in $\P^n$ such that intersecting the hyperplanes with it will yield lines that are still in general position. It sounds like a "trivial" result in linear algebra, what is the fastest way to prove this if it is true? What if I substitute $4$ with any number greater than $3$? I don't suppose $\Q$ plays any role here so we may as well substitute it with any field $K$.