The balloon-borne LSPE mission is optimized to measure the linear polarization of the Cosmic Microwave Background at large angular scales. The Short Wavelength Instrument for the Polarization Explorer (SWIPE) is composed of 3 arrays of multi-mode bolometers cooled at 0.3K , with optical components and filters cryogenically cooled below 4K to reduce the background on the detectors. Polarimetry is achieved by means of large rotating half-wave plates and wire-grid polarizers in front of the arrays. The polarization modulator is the first component of the optical chain, reducing significantly the effect of instrumental polarization. In SWIPE we trade angular resolution for sensitivity. The diameter of the entrance pupil of the refractive telescope is 45 cm, while the field optics is optimized to collect tens of modes for each detector, thus boosting the absorbed power. This approach results in a FWHM resolution of 1.8, 1.5, 1.2 degrees at 95, 145, 245 GHz respectively. The expected performance of the three channels is limited by photon noise, resulting in a final sensitivity around 0.1-0.2 muK per beam, for a 13 days survey covering 25% of the sky.