The development of next-generation data storage devices relies on the efficient control of topological spin textures at ultrafast timescales with minimal energy consumption. Here, we theoretically investigate the switching of the magnetic skyrmion in ferrimagnetic GdFeCo utilizing the helicity-independent all-optical switching (HI-AOS) driven by femtosecond laser pulses. Our study demonstrates the switching of Néel skyrmion between the two degenerate skyrmion ground states having opposite polarity and chirality. A systematic study was done by varying the laser fluence, and we found that single-shot skyrmion switching is observed for a range of fluence values, where optically induced magnetization switching is observed. The present study proves that HI-AOS is a potential mechanism for switching magnetic skyrmion at ultrafast timescales. Our results offer significant insight into implementing optical writing skyrmion-based memory devices.