I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ for some $t$. Later, he defines the orthogonal complement (within $\epsilon^{n+1}$), $\gamma^\perp$, of $\gamma_n^1$ (on page 43). Note that this is the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$ where $v$ is perpendicular to $x$. Also mentioned, the tangent bundle $\tau := T(\mathbb{R}P^n)$ of $\mathbb{R}P^n$ can be viewed as the image of the quotient map $TS^n \to T\mathbb{R}P^n$ and thus, points in $\tau$ correspond to pairs $\{(x,v), (-x,-v)\}$ where $x\cdot x = 1$ and $x \cdot v = 0$. He also proves that $\text{hom}(\gamma_n^1, \gamma^\perp) \cong \tau$. The logic is that each $(\pm x, \pm v)$ determines a map $L \to L^\perp$ (where $L$ is the line through $\pm x$), so $T(\mathbb{R}P^n)_x \cong \text{hom}(L,L^\perp)$ and thus $\text{hom}(\gamma_n^1, \gamma^\perp) \cong \tau$. This all makes lots of sense.
However, I must be missing something, since I am not able to visualize well the difference between $\tau$ and $\gamma^\perp$ (of course, they are different since their SW classes are different). I can see that when $n = 1$, $\gamma^\perp \cong \gamma_1^1$ but that $\tau$ is trivial. However, I can no longer visualize what happens for larger $n$. Part of my confusion is because I don't understand the global structure of $\gamma^\perp$. At first, I figured points would correspond to pairs $\{(x,v), (-x,-v)\}$ based on the description above, but this is just the tangent space. Perhaps points are actually of the form $\{(x,v), (-x,v)\}$? Except this is false for $n = 1$. These views implicitly assume $\gamma^\perp$ is a quotient space, which maybe is false. Does anyone have any intuitive explanations for how to think about $\gamma^\perp$, especially in comparison to $\tau$?