Using data samples collected with the BESIII detector at the BEPCII collider, we measure the Born cross section of e(+)e(-) -> p (p) over tilde at 12 center-of-mass energies from 2232.4 to 3671.0 MeV. The corresponding effective electromagnetic form factor of the proton is deduced under the assumption that the electric and magnetic form factors are equal (vertical bar G(E)vertical bar = vertical bar G(M)vertical bar). In addition, the ratio of electric to magnetic form factors, vertical bar G(E)/G(M)vertical bar, and vertical bar G(M)vertical bar are extracted by fitting the polar angle distribution of the proton for the data samples with larger statistics, namely at root s = 2232.4 and 2400.0 MeV and a combined sample at root s = 3050.0, 3060.0 and 3080.0 MeV, respectively. The measured cross sections are in agreement with recent results from BABAR, improving the overall uncertainty by about 30%. The vertical bar G(E)/G(M)vertical bar ratios are close to unity and consistent with BABAR results in the same q(2) region, which indicates the data are consistent with the assumption that vertical bar G(E)vertical bar = vertical bar G(M)vertical bar within uncertainties.