  Skyrmions are topological spin textures of interest for fundamental science
and applications. Previous theoretical studies have focused on systems with
broken bulk inversion symmetry, where skyrmions are stabilized by easy-axis
anisotropy. We investigate here systems that break surface inversion symmetry,
in addition to possible broken bulk inversion. This leads to two distinct
Dzyaloshinskii-Moriya (DM) terms with strengths $D_\perp$, arising from Rashba
spin-orbit coupling (SOC), and $D_\parallel$ from Dresselhaus SOC. We show that
skyrmions become progressively more stable with increasing
$D_\perp/D_\parallel$, extending into the regime of easy-plane anisotropy. We
find that the spin texture and topological charge density of skyrmions develops
nontrivial spatial structure, with quantized topological charge in a unit cell
given by a Chern number. Our results give a design principle for tuning Rashba
SOC and magnetic anisotropy to stabilize skyrmions in thin films, surfaces,
interfaces and bulk magnetic materials that break mirror symmetry.
