We introduce a novel procedure for computing an SVD-type approximation of a tall matrix A. Specifically, we propose a randomization-based algorithm that improves the standard Randomized Singular Value Decomposition (RSVD). Most significantly, our approach, the Row-aware RSVD (R-RSVD), explicitly constructs information from the row space of A. This leads to better approximations to Range(A) while maintaining the same computational cost. The efficacy of the R-RSVD is supported by both robust theoretical results and extensive numerical experiments. Furthermore, we present an alternative algorithm inspired by the R-RSVD, capable of achieving comparable accuracy despite utilizing only a subsample of the rows of A, resulting in a significantly reduced computational cost. This method, that we name the Subsample Row-aware RSVD (Rsub-RSVD), is supported by a weaker error bound compared to the ones we derived for the R-RSVD, but still meaningful as it ensures that the error remains under control. Additionally, numerous experiments demonstrate that the Rsub-RSVD trend is akin to the one attained by the R-RSVD when the subsampling parameter is on the order of n, for a m×n A, with m >> n. Finally, we consider the application of our schemes in two very diverse settings which share the need for the computation of singular vectors as an intermediate step: the computation of CUR decompositions by the discrete empirical interpolation method (DEIM) and the construction of reduced-order models in the Loewner framework, a data-driven technique for model reduction of dynamical systems.