Given $v$ and the inner product $\langle u,v \rangle$ of two matrices, is there any way to calculate $u$ either directly or with some additional information?
I have found an example where someone computes $u$ given the vector cross product $(u \times v)$ using the following equation. Source: http://boards.straightdope.com/sdmb/showthread.php?t=268477
$$ u = \frac{((u \cdot v)v - (u \times v) \times v)}{(v \cdot v)} $$
This assumes that you also have $(u \cdot v)$ and $(v \cdot v)$.
Is there some equivalent for the inner product? I can't seem to find anything specifically.
Not really. Counterexample with $u,v \in \mathbb{R}^2$. Let $u = (0,1)$ and $v = (1,0)$ and $w = (-\pi,0)$. Note that $u \cdot v = 0 = u \cdot w$. So given $u \cdot x = 0$ and $u = (0,1)$, there is no way to distinguish between $x=v$ or $x=w$, for example...