Can two vectors of 3-Tuples span $\mathbb R^3$? I'm just checking, but when we have $2$ vectors 
$$
V_1=\begin{pmatrix}a\\b\\c\end{pmatrix}\text{ and }V_2=\begin{pmatrix}e\\f\\g\end{pmatrix}
$$
We could theoretically span $\mathbb R^3$ real space with just these two vectors right?

While skyking gave a very elegant explanation for $\mathbb{R}^3$, there is a general fact that $n$ $n+1$ tuples can't span $\mathbb{R}^{n+1}$. One possible proof is this:
1) For $n=1$ this is obviously true.
2) Suppose any element of $\mathbb{R}^{n+1}$ can be represented as a linear combination of your tuples, then you have $e_{n+1} = (0, 0, ..., 0, 1)^T = a_1V_1+a_2V_2+...+a_nV_n$. Let $a_n \neq 0$, then $V_n = b_1V_1 + b_2V_2 + ... + b_{n-1}V_{n-1} + b_ne_{n+1}$. Thus, vectors $e_1, e_2, ..., e_n$ can be expressed using only $V_1, ..., V_{n-1}$ and $e_{n+1}$. So, ignoring the last component, you get that $n-1$ $n$-tuples span $\mathbb{R}^n$.
3) By induction, from 1) and 2) we conclude that the statement holds for all $n$.
