Solar eruptions are the most powerful drivers of space weather. To understand their cause and nature, it is crucial to know how the coronal magnetic field evolves before eruption. Here we study the formation process of a relatively large-scale magnetic flux rope (MFR) in active region NOAA~12371 that erupts with a major flare and coronal mass ejection on 2015 June 21. A data-driven numerical magnetohydrodynamic model is employed to simulate three-dimensional coronal magnetic field evolution of one-day duration before the eruption. Comparison between the observed features and our modeled magnetic field discloses how the pre-eruption MFR forms. Initially, the magnetic field lines were weakly twisted as being simple sheared arcades. Then a long MFR was formed along the polarity inversion line due to the complex photospheric motion, which is mainly shearing rather than twisting. The presence of the MFR is evidenced by a coherent set of magnetic field lines with twist number above unity. Below the MFR a current sheet is shown in the model, suggesting that tether-cutting reconnection plays a key role in the MFR formation. The MFRs flux grows as more and more field lines are twisted due to continuous injection of magnetic helicity by the photospheric motions. Meanwhile, the height of the MFRs axis increases monotonely from its formation. By an analysis of the decay index of its overlying field, we suggest that it is because the MFR runs into the torus instability regime and becomes unstable that finally triggers the eruption.
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