I learned to denote lines like following, where $\vec{p}$ is direction vector, $P$ is point in the line and < > is the notation for linear span of vector(s):
$$p: X = P + <\vec{p}>$$
This is meant to be equivalent to:
$$p: X = P + t\cdot\vec{p},\ t\in R$$
Now a plane with point $P$ and vectors $\vec{p},\vec{q}$ can be written as:
$$\rho:\ X = P + t\cdot\vec{p}+u\cdot\vec{q},\ t,u\in R$$
And I would be inclined to write it as:
$$\rho:\ X = P + <\{\vec{p}, \vec{q}\}>$$
Is that correct? I have no other reasons to use linear span than laziness (no need to define $t$ and remember not to use that letter any more) and the fact that I find it more obvious.