We study and characterize a direct route to high-dimensional chaos (i.e. not implying an intermediate low-dimensional attractor) of a system composed out of three coupled Lorenz oscillators. A geometric analysis of this medium-dimensional dynamical system is carried out through a variety of numerical quantitative and qualitative techniques, that ultimately lead to the reconstruction of the route. The main finding is that the transition is organized by a heteroclinic explosion. The observed scenario resembles the classical route to chaos via homoclinic explosion of the Lorenz model.