  We prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$
differential $k$-form with $\delta \left( a(x) \lvert du \rvert^{p-2} du
\right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n
\geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive,
bounded, Dini continuous scalar function $a$, then $du$ is continuous. This
generalizes the classical result by Stein in the scalar case and the work of
Kuusi-Mingione for the $p$-Laplacian type systems. We also discuss H\"{o}lder,
BMO and VMO regularity estimates for such systems when $p \geq 2.$
