  Given a point-line geometry P and a pappian projective space S,a veronesean
embedding of P in S is an injective map e from the point-set of P to the set of
points of S mapping the lines of P onto non-singular conics of S and such that
e(P) spans S. In this paper we study veronesean embeddings of the dual polar
space \Delta_n associated to a non-singular quadratic form q of Witt index n >=
2 in V = V(2n + 1; F). Three such embeddings are considered,namely the
Grassmann embedding gr_n,the composition vs_n of the spin (projective)
embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n;
F) and a third embedding w_n defined algebraically in the Weyl module V
(2\lambda_n),where \lambda_n is the fundamental dominant weight associated to
the n-th simple root of the root system of type Bn. We shall prove that w_n and
vs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) is
irreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is a
proper quotient of w_n. In this paper we shall study some of these submodules.
Finally we turn to universality,focusing on the case of n = 2. We prove that if
F is a finite field of odd order q > 3 then sv_2 is relatively universal. On
the contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if F
is a perfect field of characteristic 2 then vs_n is not universal,for any n>=2.
