Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write

$V = (\mathbb{C}^2, \omega)$

There are many spaces one can construct from $V$. For instance $Sym^n(V)$ is a vector space endowed with a non-degenerate complex bilinear form (respectively a non-degenerate complex skew-symmetric bilinear form) if n is even (respectively odd). Thus, we have realized $O(2k+1,\mathbb{C})$ and $Sp(2k,\mathbb{C})$ as groups of symmetry of a natural space constructed from $V$.

A more complicated example is $G_2$, which is the group of symmetries of $Sym^6(\mathbb{C}^2)$, endowed with a complex skew-symmetric trilinear form which can be defined using $\omega$ only. See for instance Theorem 1.1 (page 2) in https://arxiv.org/abs/1107.2813, though I am sure it goes back way before (not sure who is the first).

I am still struggling to get $O(2n,\mathbb{C})$ that way, as the group of symmetries of a natural space constructed from $V$, and I wonder whether the remaining exceptional Lie groups can be viewed this way.

I remark that there are more complicated spaces one can build from $V$, such as the kernel of the "symmetrization map":

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k+l}(\mathbb{C}^2)$

or, the kernel of:

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k-l}(\mathbb{C}^2)$

if $k \geq l$, the map being defined by contracting using the symplectic form $\omega$. (I believe that the right term for "natural space constructed from $V$" is a Schur functor, a term I have just met in one of Prof. Robert Bryant's answers to another post).