The detection of the diffuse gas component of the cosmic web remains a formidable challenge. In this work we study synchrotron emission from the cosmic web with simulated SKA1 observations, which can represent an fundamental probe of the warm-hot intergalactic medium. We investigate radio emission originated by relativistic electrons accelerated by shocks surrounding cosmic filaments, assuming diffusive shock acceleration and as a function of the (unknown) large-scale magnetic fields. The detection of the brightest parts of large ($>10 \rm Mpc$) filaments of the cosmic web should be within reach of the SKA1-LOW, if the magnetic field is at the level of a $\sim 10$ percent equipartition with the thermal gas, corresponding to $\sim 0.1 \mu G$ for the most massive filaments in simulations. In the course of a 2-years survey with SKA1-LOW, this will enable a first detection of the "tip of the iceberg" of the radio cosmic web, and allow for the use of the SKA as a powerful tool to study the origin of cosmic magnetism in large-scale structures. On the other hand, the SKA1-MID and SKA1-SUR seem less suited for this science case at low redshift ($z \leq 0.4$), owing to the missing short baselines and the consequent lack of signal from the large-scale brightness fluctuations associated with the filaments. In this case only very long exposures ($\sim 1000$ hr) may enable the detection of $\sim 1-2$ filament for field of view in the SKA1-SUR PAF Band1.