I am trying to prove that $\mathbb{C}^{n}$ is an irreducible representation of $O(n, \mathbb{C}) = \left\{ A \in GL(n) | A^{t}A = Id \right\}$. I presume the action is defined in the standard way, i.e. $g \cdot v = gv$ with $g \in O(n)$ and $v \in \mathbb{C}^n$. I could not think of anything different from applying the definition, thus proving there is no proper submodule of $\mathbb{C}^{n}$. Thus, by contradiction, I was trying to prove that if $A$ is a proper submodule and $v \in \mathbb{C}^n \setminus A$, then there exists an orthogonal matrix $B$ such that $Bv \notin A$. However, I don’t think I have enough information to conclude the existence of $B$. Is this even the right approach?
As a related question, once I will have proven that $O(n, \mathbb{C})$ is irreducible, I was asked to use this information to prove that $O(n, \mathbb{C})$ generates $M_n(\mathbb{C})$ as an algebra, but I don’t really know how: $O(n)$ is an algebraic linear group, so in what sense does a group generate an algebra? Is it in the sense of vector spaces in this case?