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 $V$.
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 $V$.
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).