One of the primary sources of suboptimal image quality in ultrasound imaging is phase aberration. It is caused by spatial changes in sound speed over a heterogeneous medium, which disturbs the transmitted waves and prevents coherent summation of echo signals. Obtaining non-aberrated ground truths in real-world scenarios can be extremely challenging, if not impossible. This challenge hinders the performance of deep learning-based techniques due to the domain shift between simulated and experimental data. Here, for the first time, we propose a deep learning-based method that does not require ground truth to correct the phase aberration problem and, as such, can be directly trained on real data. We train a network wherein both the input and target output are randomly aberrated radio frequency (RF) data. Moreover, we demonstrate that a conventional loss function such as mean square error is inadequate for training such a network to achieve optimal performance. Instead, we propose an adaptive mixed loss function that employs both B-mode and RF data, resulting in more efficient convergence and enhanced performance. Finally, we publicly release our dataset, comprising over 180,000 aberrated single plane-wave images (RF data), wherein phase aberrations are modeled as near-field phase screens. Although not utilized in the proposed method, each aberrated image is paired with its corresponding aberration profile and the non-aberrated version, aiming to mitigate the data scarcity problem in developing deep learning-based techniques for phase aberration correction.