La2/3Sr1/3MnO3 (LSMO) thin films of various thicknesses (6, 8, 10, 20, and 30 nm), capped by 7 nm-thick Pt layer, are grown by pulsed laser deposition on SrTiO3 (001) substrates. X-ray diffraction revealed that LSMO films are (001) oriented. Vibrating sample magnetometer is used to determine the magnetization at saturation and the magnetic dead layer thickness. This latter is around 3.4 nm, significantly thicker compared with the one induced at interfaces of Pt with ferromagnetic transition metals. Microstrip line ferromagnetic resonance (MS-FMR) is used to extract the gyromagnetic ratio, which is found to increase with LSMO thickness. MS-FMR revealed that the in-plane magnetic anisotropy is dominated by a uniaxial contribution for the Pt capped film, whereas the noncapped 10 nm-thick LSMO layer shows a fourfold anisotropy. Furthermore, the thickness dependence of the effective magnetization reveals the existence of a second-order perpendicular anisotropy term, which is thickness-dependent, and of a weak uniaxial interface anisotropy. The Gilbert damping coefficient is found to vary linearly with the inverse of the effective LSMO thickness due to spin pumping leading to relatively low spin mixing conductance of LSMO/Pt interface.