This study focuses on characterizing the channel of a reconfigurable intelligent surface (RIS)-assisted wireless system operating over Nakagami-m fading channels. Although numerous works have proposed approximate or asymptotic solutions for the RIS channel statistics (namely, the probability density function (PDF) and the cumulative distribution function (CDF)), only a limited number of studies have tackled this problem by resorting to exact approaches. Regrettably, as the number of RIS elements increases, these approaches lead to solutions that are computationally expensive or entail a high mathematical intricacy. This has prompted the analytical development undertaken in this work. Herein, our emphasis is on an exact approach. Specifically, we furnish handy and tractable formulas for the PDF and the CDF of the investigated RIS channel. The expressions introduced in this study stand out as new contributions to the literature and are arguably the most efficient exact solutions available to date. Numerical simulations revealed the heightened efficiency of our proposed PDF and CDF expressions against the state-of-the-art solutions. Furthermore, we conducted a performance assessment analysis for the considered RIS-assisted wireless communication system by deriving exact and asymptotic expressions for key performance indicators, namely the outage probability (OP) and the average bit-error rate (ABER). Comprehensive numerical simulations validated the accuracy of our analytical results.