Many dynamical effects in biology, social and technological complex systems have recently revealed their relevance to group interactions beyond traditional dyadic relationships between individual units. In this paper, we propose a growing simplicial network to model the higher-order interactions represented by clique structures. We analytically study the degree distribution and clique distribution of the network model. As an important degree-based topological index, Sombor index of the model has been derived in an iterative manner and an approximation method with closed expression is proposed. Moreover, we observe power-law and small-word effect for the simplicial networks and examine the effectiveness of the approximation method for Sombor index through computational experiments. We discover the scaling constant for Sombor index with the evolution of the network when the initial seed network is modeled as an Erdős-Rényi random graph. Our findings suggest the relevance and potential applicability of simplicial networks in modelling higher-order interactions in complex networked systems.