In this work, a new second-order conic optimization model for the Euclidean Steiner tree problem in d-space, where the dimension d is at least 3, will be presented. Also, two strategies for eliminating isomorphic trees will be developed. Computational results highlighting the efficiency of this model compared to existing models for the Euclidean Steiner Tree problem will be discussed. Finally, enforcing the proposed model with any of the isomorphic tree elimination strategies permits us to compute for the first time, to the best of our knowledge, the minimum Steiner tree for the cube in R^3 .