One of the key advancements in wireless communication, particularly in 5G and 6G networks, is cell-free massive multiple-input multiple-output (CF M-MIMO). This technology uses a large number of distributed antennas to serve multiple users simultaneously, providing higher spectral efficiency (SE), improved coverage, and better interference control compared to traditional cellular networks. However, achieving efficient channel estimation with low computational complexity remains a challenge. Several algorithms have been developed to address these challenges, with the phase-aware minimum mean square error (PA-MMSE) estimator standing out as a high-performance option. Although effective, the PA-MMSE estimator is limited by its high computational complexity. To overcome these challenges, this paper introduces a phase-aware element-wise MMSE (PA-EW-MMSE) estimator, which incorporates QR decomposition (where Q is an orthogonal matrix and R is an upper triangular matrix) along with a user-side precoding matrix. The proposed estimator is evaluated in terms of uplink (UL) SE using MMSE combining. Additionally, energy efficiency (EE) and area throughput are calculated from SE. Simulation results demonstrate that the proposed PA-EW-MMSE estimator significantly reduces computational complexity while delivering superior SE, EE, and area throughput compared to existing methods.