As generalizations of random graphs, random simplicial complexes have been receiving growing attention in the literature. In this paper, we naturally extend the Random Connection Model (RCM), a random graph that has been extensively studied for over three decades, to a random simplicial complex, recovering many models currently found in the literature as special cases. In this new model, we derive quantitative central limit theorems for a generalized Euler characteristic in various asymptotic scenarios. We will accomplish this within a very general framework, where the vertices of the simplicial complex are drawn from an arbitrary Borel space.