  We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke
algebras $\mathcal{R}^\Lambda_n$ of type $A$ when $e=0$ (the linear quiver) or
$e\ge n$. We show that these algebras are quasi-hereditary graded cellular
algebras by giving explicit homogeneous bases for them. When $e=0$ we show that
the KLR grading on the quiver Hecke algebras is compatible with the gradings on
parabolic category $\mathcal{O}$ previously introduced in the works of
Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that
when $e=0$ our graded Schur algebras are Koszul over field of characteristic
zero. Finally, we give an LLT-like algorithm for computing the graded
decomposition numbers of the quiver Schur algebras in characteristic zero when
$e=0$.
