Does the span of a set of a vectors set always cover the whole space?

Question

Suppose we have $V= \{v_{1},v_{2},...,v_{p}\},v_{i} \in R^{n}$ vectors set .
Span of $V$ is the set of all possible linear combinations of $v_{1},v_{2},v_{3}, ,…,v_{p}$ for some $C= \{c_{1}, c_{2}, ..., c_{p}\}$ scalars.
Questions:
If we don't fix the scalars $C= \{c_{1}, c_{2}, ..., c_{p}\}$, and let them to be any value from $R$ - $c_{i} \in R$, does the span covers all the $R^{n}$ space ? (for 2d - the whole 2d plane) ?
And if we fix the $C$ set, does the span always plots a line in $R^n$ ?
Can we have an example say for $R^2$ case, when a span plots a plane in 2d (doesn't cover the whole 2d plane) ?
Supposing all the vectors don't lie on the same line.