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An angular analysis of the decay B[over ¯]→D^{*}ℓ^{-}ν[over ¯]_{ℓ}, ℓ∈{e,μ}, is reported using the full e^{+}e^{-} collision data set collected by the BABAR experiment at the ϒ(4S) resonance. One B meson from the ϒ(4S)→BB[over ¯] decay is fully reconstructed in a hadronic decay mode, which constrains the kinematics and provides a determination of the neutrino momentum vector. The kinematics of the semileptonic decay is described by the dilepton mass squared, q^{2}, and three angles. The first unbinned fit to the full four-dimensional decay rate in the standard model is performed in the so-called Boyd-Grinstein-Lebed approach, which employs a generic q^{2} parametrization of the underlying form factors based on crossing symmetry, analyticity, and QCD dispersion relations for the amplitudes. A fit using the more model-dependent Caprini-Lellouch-Neubert (CLN) approach is performed as well. Our form factor shapes show deviations from previous fits based on the CLN parametrization. The latest form factors also provide an updated prediction for the branching fraction ratio R(D^{*})≡B(B[over ¯]→D^{*}τ^{-}ν[over ¯]_{τ})/B(B[over ¯]→D^{*}ℓ^{-}ν[over ¯]_{ℓ})=0.253±0.005. Finally, using the well-measured branching fraction for the B[over ¯]→D^{*}ℓ^{-}ν[over ¯]_{ℓ} decay, a value of |V_{cb}|=(38.36±0.90)×10^{-3} is obtained that is consistent with the current world average for exclusive B[over ¯]→D^{(*)}ℓ^{-}ν[over ¯]_{ℓ} decays and remains in tension with the determination from inclusive semileptonic B decays to final states with charm.