The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.