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In this paper we define an explicit basis for the $\mathfrak{gl}_n$-webalgebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arcalgebra) using categorified $q$-skew Howe duality.  Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' gradedcellular basis and has two major applications: it gives rise to an explicitisomorphism between a certain idempotent truncation of a thick calculuscyclotomic KLR algebra and $H_n(\vec{k})$, and it gives an explicit gradedcellular basis of the $2$-hom space between two $\mathfrak{gl}_n$-webs. We usethis to give a (in principle) computable version of colored Khovanov-Rozansky$\mathfrak{gl}_n$-link homology, obtained from a complex defined purelycombinatorially via the (thick cyclotomic) KLR algebra and needs only $F$.

$\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies