The following argument suggests that the factor $O(n)$ cannot be improved. Suppose for a contradiction that $x\in\{0,1\}^n$ and after queries with $v_1,\dots,v_{n-1}\in\mathbb R^n$ we determined that $x=(1,\dots,1)$. Pick an index $j\in\{1,\dots,n\}$ which was an answer for no queries and let $x'\in\{0,1\}^n$ be the same as $x$ with the only difference that the $j$th component of $x'$ is $0$. Then for $x'$ we obtain the same query answers as for $x$, so we cannot to distinguish $x$ from $x'$, a contradiction.

The following argument suggests that the factor $O(n)$ cannot be improved. Suppose for a contradiction that $x\in\{0,1\}^n$ and after queries with $v_1,\dots,v_{n-1}\in\mathbb R^n$ we determined that $x=(1,\dots,1)$. 

Pick an index $j\in\{1,\dots,n\}$ which was an answer for no queries and let $x'\in\{0,1\}^n$ be the same as $x$ with the only difference that the $j$th component of $x'$ is $0$. Then for $x'$ we obtain the same query answers as for $x$, so we cannot hope to distinguish $x$ from $x'$, a contradiction.