1. We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space $V^{\mathbb{R}}:={\mathbb{R}}^n$ (where $n=p+q$).
By definition,
$$
{\bf O}(p,q)={\rm Aut}({\mathbb{R}}^n,F^{\mathbb{R}}_{p,q}).
$$
By Serre, Local Fields, Section X.2, Corollary 1 of Proposition 4, page 153 in the English version, there is a canonical bijection between
$H^1({\mathbb{R}},{\bf O}(p,q))$ and the set of isomorphism classes of nondegenerate quadratic forms of $n$ variables over ${\mathbb{R}}$,
i.e. isomorphism classes of pairs $(V,F)$ over ${\mathbb{R}}$
where $V$ is an $n$-dimensional vector space and $F$ is a nondegenerate quadratic form on $V$.

2. Similarly,
we denote by $F^{\mathbb{C}}_{p,q}$ the Hermitian ${\mathbb{C}}/{\mathbb{R}}$ form
$$
F^{\mathbb{C}}_{p,q}(x)=x_1 \bar x_1+\dots+x_p\bar x_p-(x_{p+1}\bar x_{p+1}+\dots+x_{p+q}\bar x_{p+q})
$$
on the vector space $V^{\mathbb{C}}:={\mathbb{C}}^n$ (where $n=p+q$).
By definition,
$$
{\bf U}(p,q)={\rm Aut}({\mathbb{C}}^n,F^{\mathbb{C}}_{p,q}).
$$
Then there is a canonical bijection between
$H^1({\mathbb{R}},{\bf U}(p,q))$ and the set of isomorphism classes of nondegenerate Hermitian ${\mathbb{C}}/{\mathbb{R}}$ forms of $n$ variables,
i.e. isomorphism classes of pairs $(V,F)$ over ${\mathbb{C}}$
where $V$ is an $n$-dimensional vector space and $F$ is a nondegenerate Hermitian form on $V$.

3. Similarly we denote by $F^{\mathbb{H}}_{p,q}$ the Hermitian ${\mathbb{H}}/{\mathbb{R}}$ form  on the vector space $V^{\mathbb{H}}:={\mathbb{H}}^n$ given by the same formula
(where $n=p+q$ and by ${\mathbb{H}}$ we denote the skew field of Hamilton's quaternions with the canonical involution).
By definition,
$$
{\bf Sp}(p,q)={\rm Aut}({\mathbb{H}}^n,F^{\mathbb{H}}_{p,q}).
$$
Then there is a canonical bijection between
$H^1({\mathbb{R}},{\bf Sp}(p,q))$ and the set of isomorphism classes of nondegenerate Hermitian ${\mathbb{H}}/{\mathbb{R}}$ forms of $n$ variables.


> **Question 1.**
What is classified by $H^1({\mathbb{R}},{\bf SO}(p,q))$?


> **Question 2.**
What is classified by $H^1({\mathbb{R}},{\bf SU}(p,q))$?


I need explicit functorial descriptions as above rather than formulas for the cardinalities of the corresponding sets.
My feeling is that in both questions $H^1$ classifies nondegenerate forms of n variables with the same determinant as $F_{p,q}$
(modulo squares in Question 1 and modulo norms in Question 2).