The bundle classified by this element of $\pi_{n-1}SO(n)$ is the tangent bundle of $S^n$, so the image under $\theta$ is twice a generator if $n$ is even, zero if $n$ is odd.

To say what $\theta$ is explicitly in the case $n=4$ in terms of the $\mathbb Z$-basis for $\pi_3(SO(4)$, we would have to know what basis you have in mind, but for the one I usually think of it takes both basis elements to the same generator.

Edit: Note also that the map $\theta:\pi_{n-1}SO(n)\to H^n(S^n)$ is isomorphic to the map $\theta:\pi_{n-1}SO(n)\to \pi_{n-1}(S^{n-1})$ induced by the bundle projection $SO(n)\to S^{n-1}$.

The bundle classified by this element of $\pi_{n-1}SO(n)$ is the tangent bundle of $S^n$, so the image under $\theta$ is twice a generator if $n$ is even, zero if $n$ is odd.

To say what $\theta$ is explicitly in the case $n=4$ in terms of the $\mathbb Z$-basis for $\pi_3SO(4)$, we would have to know what basis you have in mind, but for the one I usually think of it takes both basis elements to the same generator.

Edit: Note also that the map $\theta:\pi_{n-1}SO(n)\to H^n(S^n)$ is isomorphic to the map $\theta:\pi_{n-1}SO(n)\to \pi_{n-1}(S^{n-1})$ induced by the bundle projection $SO(n)\to S^{n-1}$.

The bundle classified by this element of $\pi_{n-1}SO(n)$ is the tangent bundle of $S^n$, so the image under $\theta$ is twice a generator if $n$ is even, zero if $n$ is odd.

To say what $\theta$ is explicitly in the case $n=4$ in terms of the $\mathbb Z$-basis for $\pi_3(SO(4)$, we would have to know what basis you have in mind, but for the one I usually think of it takes both basis elements to the same generator.

The bundle classified by this element of $\pi_{n-1}SO(n)$ is the tangent bundle of $S^n$, so the image under $\theta$ is twice a generator if $n$ is even, zero if $n$ is odd.

To say what $\theta$ is explicitly in the case $n=4$ in terms of the $\mathbb Z$-basis for $\pi_3(SO(4)$, we would have to know what basis you have in mind, but for the one I usually think of it takes both basis elements to the same generator.

Edit: Note also that the map $\theta:\pi_{n-1}SO(n)\to H^n(S^n)$ is isomorphic to the map $\theta:\pi_{n-1}SO(n)\to \pi_{n-1}(S^{n-1})$ induced by the bundle projection $SO(n)\to S^{n-1}$.