The answer is *no*. 

In fact, let $\Gamma \subset \mathbb{P}^3$ be  the set  of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be contained in a Veronese surface.

Another characterization is the following: the set $\Gamma$ arises as a linear section of the secant variety $\textrm{Sec}(C_6)$, where $C_6 \subset \mathbb{P}^6$ is a rational normal curve. Note that $\textrm{Sec}(C_6)$ has precisely degree $10$, since $10$ is the number of nodes of a general projection of $C_6$ to $\mathbb{P}^2$.

For (many) more details, see

D. Eisenbud, S. Popescu: [The projective geometry of the Gale transform][1], *Journal of  Algebra* **230** (2000), 127-173,

in particular Example 6.3 p. 153.   
 


  [1]: http://www.sciencedirect.com/science/article/pii/S0021869399979401