  We introduce the notion of intrinsic Grassmannians which generalizes the well
known weighted Grassmannians. An intrinsic Grassmannian is a normal projective
variety whose Cox ring is defined by the Pl\"ucker ideal $I_{d,n}$ of the
Grassmannian $\mathrm{Gr}(d,n)$. We give a complete classification of all
smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and
prove an explicit formula to compute the total number of such varieties for an
arbitrary $n$. We study their geometry and show that they satisfy Fujita's
freeness conjecture.
