We present a full-field technique that allows label-free cytoskeletal network imaging inside living cells. This noninvasive technique allows monitoring of the cytoskeleton dynamics as well as interactions between the latter and organelles on any timescale. It is based on high-resolution quantitative phase imaging (modified Quadriwave lateral shearing interferometry) and can be directly implemented using any optical microscope without modification. We demonstrate the capability of our setup on fixed and living Chinese hamster ovary cells, showing the cytoskeleton dynamics in lamellipodia during protrusion and mitochondria displacement along the cytoskeletal network. In addition, using the quantitative function of the technique, along with simulation tools, we determined the refractive index of a single tubulin microtubule to be n_tubu = 2.36 +/- 0.6 at Lambda = 527 nm.