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Magnetic skyrmions are topologically protected states that are very promising for the design of the next generation of ultra-low-power electronic devices. In this letter, we propose a magnetic tunnel junction based spin-transfer torque diode with a magnetic skyrmion as ground state and a perpendicular polarizer patterned as nano-contact for a local injection of the current. The key result is the possibility to achieve sensitivities (i.e., detection voltage over input microwave power) larger than 2000 V/W for optimized contact diameters. We also pointed out that large enough voltage controlled magnetocrystalline anisotropy could significantly improve the sensitivity. Our results can be very useful for the identification of a class of spin-torque diodes with a non-uniform ground state and to understand the fundamental physics of the skyrmion dynamical properties.