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 a zero dimensional scheme of length $10$ given by the simple intersection points of two Veronese surfaces, and this condition is not satisfied by a collection of $10$ general points.

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

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

in particular Example 6.3 p. 153.

**(2)** D. Eisenbud, K. Hulek and S. Popescu: A note on the intersection of Veronese surfaces, *Commutative Algebra, Singularities and Computer Algebra*, Volume **115** of the series NATO Science Series (2003), pp 127-139,

in particular Section 3.