Global sensitivity analysis (GSA) aims at quantifying the contribution of input variables over the variability of model outputs. In the frame of functional outputs, a common goal is to compute sensitivity maps (SM), i.e sensitivity indices at each output dimension (e.g. time step for time series, or pixels for spatial outputs). In specific settings, some works have shown that the computation of Sobol' SM can be speeded up by using basis expansions employed for dimension reduction. However, how to efficiently compute such SM in a general setting has not received too much attention in the GSA literature. In this work, we propose fast computations of Sobol' SM using a general basis expansion, with a focus on statistical estimation. First, we write a closed-form expression of SM in function of the matrix-valued Sobol' index of the vector of basis coefficients. Secondly, we consider pick-freeze (PF) estimators, which have nice statistical properties (in terms of asymptotical efficiency) for Sobol' indices of any order. We provide similar basis-derived formulas for the PF estimator of Sobol' SM in function of the matrix-valued PF estimator of the vector of basis coefficients. We give the computational cost, and show that, compared to a dimension-wise approach, the computational gain is substantial and allows to calculate both SM and their associated bootstrap confidence bounds in a reasonable time. Finally, we illustrate the whole methodology on an analytical test case and on an application in non-Newtonian hydraulics, modelling an idealized dam-break flow.