In this paper, we classify all generalized quadrangles weakly embedded of degree~$2$ in projective space. More exactly, given a (possibly infinite) generalized quadrangle $\Gamma=(\cP,\cL,\op)$ and a map $\pi$ from $\cP$ (respectively $\cL$) to the set of points (respectively lines) of a projective space $\PG(V)$, $V$ a vector space over some skew field (not necessarily finite-dimensional), such that: \begin{itemize} \item[(i)] $\pi$ is injective on points, \item[(ii)] if $x\in\cP$ and $L\in\cL$ with $x\op L$, then $x^\pi$ is incident with $L^\pi$ in $\PG(V)$, \item[(iii)] the set of points $\{x^\pi \mid x\in \cP\}$ generates $\PG(V)$, \item[(iv)] if $x,y \in \cP$ such that $y^\pi$ is contained in the subspace of $\PG(V)$ generated by the set $\{z^\pi \mid z$ is collinear with $x$ in $\Gamma \}$, then $y$ is collinear with $x$ in $\Gamma$, \item[(v)] there exists a line of $\PG(V)$ not in the image of $\pi$ and which meets $\mathcal{P}^\pi$ in precisely $2$ points, \end{itemize} then we show that $\Gamma$ is a Moufang quadrangle and we can explicitly describe the weak embedding of $\Gamma$ in $\PG(V)$. This completes the classification of all weak embeddings of arbitrary generalized quadrangles (using the classification of \linebreak Moufang quadrangles).