In the book Finite Model Theory by Ebbinghaus-Flum (and in other sources), it is shown that if $\tau$ is any finite vocabulary consisting of just relation and constant symbols, and if $\mathcal{A}$ is any finite $\tau$-structure with $\overline{a} \in A$, then for every $m \geq 0$, there is a so-called first-order $m$-Hintikka formula $\varphi_{\mathcal{A}, \overline{a}}^m[\overline{x}]$ (in $length(\overline{a})$ free variables) that characterizes the $m$-isomorphism type of $\mathcal{A}, \overline{a}$, in the sense that if $\mathcal{B}$ is any finite $\tau$-structure with $\overline{b} \in B$, then $$\mathcal{B} \models \varphi_{\mathcal{A}, \overline{a}}^m[\overline{b}]$$ iff $\overline{a}$ and $\overline{b}$ satisfy exactly the same first-order $\tau$-formulas of quantifier rank $\leq m$ in $\mathcal{A}$ and $\mathcal{B}$ respectively.
Now, when they move from discussing first-order logic to the infinitary first-order logics $\mathcal{L}_{\infty, \omega}$ and $\mathcal{L}_{\omega_1, \omega}$, they do not define analogues of these Hintikka formulas. So my question is, given a finite vocabulary $\tau$ as above, and a finite $\tau$-structure $\mathcal{A}$ and tuple $\overline{a} \in A$, is it possible to define an 'infinitary' Hintikka formula $\varphi_{\mathcal{A}, \overline{a}}^\infty[\overline{x}]$ that characterizes the '$\infty$-isomorphism type' of $\mathcal{A}, \overline{a}$, in the sense that if $\mathcal{B}$ is any finite $\tau$-structure with $\overline{b} \in B$, then $$\mathcal{B} \models \varphi_{\mathcal{A}, \overline{a}}^\infty[\overline{b}]$$ iff $\overline{a}$ and $\overline{b}$ satisfy exactly the same $\tau$-formulas of $\mathcal{L}_{\infty, \omega}$ in $\mathcal{A}$ and $\mathcal{B}$ respectively?