Given a non-singular quadratic form q of maximal Witt index on \(V := V(2n+1,\mathbb{F})\), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ k be the k-grassmannian of Δ. Let e k be the embedding of Δ k into PG(⋀ k V) mapping every point 〈v 1,v 2,…,v k 〉 of Δ k to the point 〈v 1∧v 2∧⋯∧v k 〉 of PG(⋀ k V). It is known that if \(\mathrm{char}(\mathbb{F})\neq2\) then \(\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}\). In this paper we give a new very easy proof of this fact. We also prove that if \(\mathrm{char}(\mathbb{F}) = 2\) then \(\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}\). As a consequence, when 1 2 or a number field, n>k and k=2 or 3, then e k is universal.