Well, two (linearly independent) vectors determine a plane...but the kind of plane they determine is special, because, being a vector space it has to pass through the 0-vector.
So, for example, we have $P_0 = (0,0,0)$ for any planar subspace. So that means we can write it as:
$Ax + By + Cz = 0$ (since $P_0 = (x_0,y_0,z_0) = (0,0,0)$).
Since we have $U$ contained in our plane, we have that any vector of the form $(0,0,z)$ is in our plane, so this gives us (using $z = 1$):
$A0 + B0 + C = 0$ which forces $C = 0$.
So our plane has the equation:
$Ax + By = 0$, which is the equation for a line $L$ in the $xy$-plane, and our plane, is of the form:
$\{(x,y,z) \in \Bbb R^3: (x,y) \in L, z \in \Bbb R\}$.
If our plane is non-degenerate (that is $A$ and $B$ are not BOTH zero), then we can write either:
$y = -(B/A)x$, and our points in our plane are of the form:
$(x,-(B/A)x,z)$ which is spanned by the vectors $\{(1,-B/A,0),(0,0,1)\}$, or:
$x = -(A/B)y$ and our plane points have the form $(-(A/B)y,y,z)$ which is spanned by $\{(-A/B,1,0),(0,0,1)\}$.
But let's find two LI vectors such that $\mathbf{n}\cdot \vec{P} = 0$.
Since $(0,0,1)$ is in our plane, if $\mathbf{n} = (n_1,n_2,n_3)$, the condition:
$\mathbf{n}\cdot \vec{P} = 0$ gives $n_3 = 0$.
I claim we can pick $(x,y,z)$ such that $(x,y,z)\cdot (n_1,n_2,0) = 0$ and $\{(x,y,z),(0,0,1)\}$ are LI. Furthermore, you can see by inspection that it makes no difference what $z$ is (since $n_3 = 0$) so we'll make it easy on ourselves by setting $z = 0$.
We will assume $\mathbf{n} \neq (0,0,0)$ (usually $\mathbf{n}$ is chosen to be a unit vector, to avoid this troublesome possibility-everything is perpendicular to the origin).
Let's suppose $n_2 \neq 0$. (I leave the case $n_1 \neq 0$ to you). Chose $y = -\dfrac{n_1}{n_2}x$. This satisfies our requirements. It is not hard to see that:
$\left\{\left(1,-\dfrac{n_1}{n_2},0\right),(0,0,1)\right\}$ are LI, and we are done.
Conversely, it is clear that $\{(x,y,0),(0,0,1)\}$ is LI, provided not both $x$ and $y$ are $0$. Set:
$W = \text{span}(\{(x,y,0),(0,0,1)\})$. Let's suppose $x \neq 0$ (you can do the $y \neq 0$ case).
Set $\mathbf{u} = \left(\dfrac{y}{x},-1,0\right)$ and take $\mathbf{n} = \dfrac{\mathbf{u}}{\|\mathbf{u}\|}$.
If $\mathbf{w} \in W$, we have:
$\mathbf{w}\cdot\mathbf{n} = \dfrac{1}{\sqrt{\dfrac{y^2}{x^2}+1}}(ax,ay,bz)\cdot\left(\dfrac{y}{x},-1,0\right)$
$= \dfrac{1}{\sqrt{\dfrac{y^2}{x^2}+1}}(ay - ay + 0) = 0$.
On the other hand if $\mathbf{v}\cdot\mathbf{n} = 0$, then writing $\mathbf{v} = (a,b,c) = (a,b,0) + (0,0,c)$ leads us to conclude:
$a\dfrac{y}{x} - b = 0$ (since $\mathbf{v}\cdot\mathbf{n} =0 \implies \mathbf{v}\cdot\mathbf{u} =0)$, that is:
$b = a\dfrac{y}{x}$, so that $(a,b,0) = \left(a,a\dfrac{y}{x},0\right) = \dfrac{a}{x}(x,y,0)$, thus:
$\mathbf{v} = (a,b,c) = \dfrac{a}{x}(x,y,0) + c(0,0,1) \in W$.
I'll address your question about uniqueness in another post.