For cell complexes${}^1$ $X$ we have an isomorphism

$$ K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}), $$

which is induced by the Chern character.

What is the analogous statement for $KO(X)$?

${}^1$:Hatcher states finite, but I've seen arbitrary CW-complexes stated as well.

For cell complexes${}^1$ $X$ we have an isomorphism

$$ K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}), $$

which is induced by the Chern character.

What is the analogous statement for $KO(X)$?

${}^1$:Hatcher states finite, but I've seen arbitrary CW-complexes stated as well.

edit: The footnote seems wrong, as by the comments.