We provide a geometric characterization of $k$-dimensional $\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when $k=2$, they are one-weight, as in the Hamming-metric case. We then focus on constant rank-profile codes in the sum-rank metric, which are a special family of one weight-codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the $n$-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of $\mathrm{GL}(k,q^m)$. They turn out to be constant rank-profile - and hence one-weight - and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on $2$-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when $q=2$.