The aim of the SAMADHA (South Atlantic Magnetic Anomaly Dosimetry at High Altitude) project is to monitor the spectrum of the secondary cosmic ray neutrons and the environmental dose due to cosmic rays at very high altitude in the South Atlantic Anomaly (SAA) region. In the SAA the confinement volume of the inner Van Allen belt, containing magnetically trapped protons with energy up to a few GeV, has its minimum distance from the Earth surface of approximately 200 km. SAMADHA studies the possible variations of the neutron flux and relative dose rate at ground level due to perturbations of the inner Van Allen belt during geomagnetic storms. The measurement site is the Cosmic ray Laboratory at Chacaltaya (5240 m) in Bolivia, where a Bonner sphere neutron spectrometer and dosimetric instruments are in operation since winter 2022-2023, aimed to monitor the neutron flux during the phase of increasing solar activity of cycle 25. The paper reports the first results of the experiment.