Let $f(u)$ and $g(v)$ be two polynomials of degree $k$ and $\ell$ respectively, not both linear, which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),\\uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study $ψ$-skew cyclic and $θ_t$-skew constacyclic codes over the ring $\mathcal{R}$ where $ψ$ and $θ_t$ are two automorphisms defined on $\mathcal{R}$.