Two distinct planes in a projective space intersect in a line?

Question

I see from another post 'Can two planes intersect in a point?' that planes in Euclidean spaces of dimension 4 or more can intersect in a single point. How does this work in projective spaces? Can 2 distinct non-skew planes intersect in a point? Since any projective space that contains such planes necessarily has dimension at least 4, and is thus Desarguesian, it seems to me that such planes can exist if and only if there exist 2 3-dimensional subspaces of a 5-dimensional vector space (over a division ring) that intersect in a line. I have a mental block on this problem - any thoughts? Thanks