Given a subspace $V\subseteq \Bbb R^n$. Let $P$ be the orthogonal projector matrix onto $V$ (where we consider $\Bbb R^n$ with the standard inner product). Let ${\bf e}_i:=(0,...,1,...,0)$ be the $i$-th canonical unit vector. The angle between $V$ and $\mathbf e_i$ can be computed via
$$\alpha_i:=\arccos\|P{\bf e}_i\|.$$
Question: Is there a way to reconstruct $V$ from these angles $\alpha_i$?
Obviously, $V$ can only be reconstructed up to reflections on the coordinate planes $\langle\mathbf e_i,\cdot\rangle=0$. Note that also the dimension of $V$ is unknown and needs to be reconstructed.
Update
As pointed out by Hagen von Eitzen in an answer, dimension considerations show that reconstruction only works for $\dim(V)=1$ or $\dim(V)=n-1$.
Therefore (and because I do not know the dimension of $V$), I ask whether at least $\dim(V)$ is reconstructible in general.
If $\dim V=n-1$ (and we know that), then yes: There exists a unit vector $v$ with $V=\{v\}^\perp$ and we have $Px=x-\langle x,v\rangle v$, thus $\|Pe_i\|^2=1-\langle e_i,v\rangle^2 $, and find all components of $v$ up to sign (i.e., $v$ and ultimately $V$ up to reflections).
If $k=n-2$, for example, the set of $k$-dimensional subspaces of $V$ is a manifold of dimension $(n-1)(n-2)$ and this is $>n$ for $n>3$, hence the continuous map $U\mapsto (\|P_Ue_1\|,\ldots, \|P_Ue_n\|)\in \Bbb R^n$ cannot be locally invertible.