  We construct a supercategory that can be seen as a skew version of
(thickened) KLR algebras for the type $A$ quiver. We use our supercategory to
construct homological invariants of tangles and show that for every link our
invariant gives a link homology theory supercategorifying the Jones polynomial.
Our homology is distinct from even Khovanov homology and we present evidence
supporting the conjecture that it is isomorphic to odd Khovanov homology. We
also show that cyclotomic quotients of our supercategory give
supercategorifications of irreducible finite-dimensional representations of
$\mathfrak{gl}_n$ of level 2.
