To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of Mn(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra Mn(K). Here, we strengthen this statement in three ways: we show that Mn(K) is spanned by the products of the form AB with (A,B)∈V2; we prove that every matrix in Mn(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of Mn(K) and n>2, we show that every matrix in Mn(K) is a product of two elements of V.