Skyrmion crystals have been primarily discovered under a magnetic field for materials with non-centrosymmetric interactions. More recent developments have investigated the stability of skyrmion crystals in itinerant magnets without magnetic field. In this study, we find that a type of skyrmion crystal with two topological charges per unit cell and no magnetization at the ferromagnetic point in reciprocal space, SkX-2, is naturally stabilized in an $SO(3)$-symmetric model with short-range interactions realized by the Heisenberg model on the triangular lattice with scalar chirality. We complement our numerical results with a theoretical analysis that quantitatively describes the transition from the ferromagnetic ground state to the SkX-2 and the evolution of the topological charge density. Despite the constraints given by the Mermin-Wagner theorem at finite temperature, the SkX-2 can either exhibit a first-order phase transition with translation symmetry breaking, or a continuous transition to a floating solid depending on the charge density which is given by the strength of the scalar chirality interaction. Finally, the numerical results indicate the presence of ${\mathbb{Z}}_2$-vortices at finite temperature above the tetrahedral order in the antiferromagnetic region of our model, suggesting the existence of a vortex topological transition similar to that reported in the ordering of the pure antiferromagnetic model.