Why can the Orthogonal group be split up in this way? In my groups course at university, we’ve spent a while on the orthogonal group, leading up the the conclusion that
$$
  \mathrm{O}_n
  =
  \mathrm{SO}_n
  \mathbin{\dot{\cup}}
  \begin{pmatrix}
    -1 &   &        &   \\
       & 1 &        &   \\
       &   & \ddots &   \\
       &   &        & 1
  \end{pmatrix}
  \mathrm{SO}_n \,.
$$
The proof given was just “because cosets partition”.
I just want to know: Is this true because of a specific choice of matrix or will any matrix in $\mathrm{O}_n \setminus \mathrm{SO}_n$ do? How do we know there arent any other cosets? (I.e., that this expression covers everything in $\mathrm{O}_n$).
 A: The subgroup $\mathrm{SO}(n)$ of $\mathrm{O}(n)$ has index $2$.
Indeed, $\mathrm{SO}(n)$ is the kernel of the surjective group homomorphism
$$
  \det \colon \mathrm{O}(n) \longrightarrow \{ 1, -1 \} \,,
$$
so
$$
  [\mathrm{O}(n) : \mathrm{SO}(n)]
  = | \mathrm{O}(n) / \mathrm{SO}(n) |
  = | \{1, -1\} |
  = 2 \,.
$$
Let more generally $G$ be a group and $H$ a subgroup of $G$ of index $2$.
The group $G$ is the disjoint union of the left cosets with respect to $H$, and there are precisely $2$ such cosests.
(Because the number of cosets is given by $[G : H] = 2$.)
One of these cosets is $H$ itself, whence
$$
  G = H \mathbin{\dot{\cup}} gH
$$
for every element $g$ of $G$ with $gH ≠ H$.
We have $gH = H$ if and only if $g ∈ H$, so any element $g$ of $G$ with $g ∉ H$ does the trick.
To explicitly answer your questions:

Is this true because of a specific choice of matrix or will any matrix in $\mathrm{O}_n \setminus \mathrm{SO}_n$ do?

Yes, any matrix of $\mathrm{O}(n)$ that is not contained in $\mathrm{SO}(n)$ will do the trick.

How do we know there arent any other cosets? (I.e., that this expression covers everything in $\mathrm{O}_n$).

There are only two cosets because $\mathrm{SO}(n)$ has index $2$ in $\mathrm{O}(n)$.
This in turn is true because $\mathrm{SO}(n)$ is the kernel of a surjective homomorphism from $\mathrm{O}(n)$ into a group of order $2$.
