We address the problem of establishing exponential stability for parameterized families of linear time-varying systems, uniformly in ‘the’ parameter. Our contribution is specifically for "skew-symmetric" systems (the use of quotes " " is motivated by having one non-zero element in the main diagonal). Our main analysis tool is a reformulation for families of systems, of the well known in (adaptive control theory) concept of persistency of excitation. However, our proof is based on modern results which can be interpreted as an "integral" version of Lyapunov theorems; rather than on the concept of uniform complete observability which is most common in the literature of linear adaptive control systems. As a byproduct we also provide a result for uniform exponential stability for nonlinear systems under the assumption that we know a Lyapunov function with negative semidefinite derivative.