This letter proposes highly accurate closed-form approximations to channel distributions of two different reconfigurable intelligent surface (RIS)-based wireless system setups, namely, dual-hop RIS-aided (RIS-DH) scheme and RIS-aided transmit (RIS-T) scheme. Differently from previous works, the proposed approximations reveal to be very tight for arbitrary number {N} of reflecting metasurface's elements. Our findings are then applied to the performance analysis of the considered systems, in which the outage probability, bit error rate, and average channel capacity are derived. Results show that the achievable diversity orders G_{d} for RIS-DH and RIS-T schemes are N-1< G_{d}< N and {N} , respectively. Furthermore, it is revealed that both schemes can not provide the multiplexing gain and only diversity gains are achieved. For the RIS-DH scheme, the channels are similar to the keyhole multiple-input multiple-output (MIMO) channels with only one degree of freedom, while the RIS-T scheme is like the transmit diversity structure.