We propose a variation of Construction A of lattices from linear codes defined using the quotient $Λ/\mathfrak pΛ$ of some order $Λ$ inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $Λ/\mathfrak pΛ$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.