  To match the precision of present and future measurements of W-boson
production at hadron colliders, electroweak radiative corrections must be
included in the theory predictions. In this paper we consider their effect on
the transverse momentum (p_T) distribution of W bosons, with emphasis on large
p_T. We evaluate the full electroweak O(\alpha) corrections to the process pp
-> Wj including virtual and real photonic contributions. We also provide
compact approximate expressions which are valid in the high-energy region,
where the electroweak corrections are strongly enhanced by logarithms of
\hat{s}/M_W^2. These expressions include quadratic and single logarithms at one
loop as well as quartic and triple logarithms at two loops. Numerical results
are presented for proton-proton collisions at 14 TeV. The corrections are
negative and their size increases with p_T. At the LHC, where transverse
momenta of 2 TeV or more can be reached, the one- and two-loop corrections
amount up to -40% and +10%, respectively.
