The dual space $V^*$ of a vector space $V$ over $F$ is defined as $V\to F$.
This seems like a weird definition. In my limited experience with linear algebra, I've thought of the dual vector of a vector as the row vector version of that (column) vector.
What is the rationale for the dual space concept?
This definition is much more general than your suggestion. When $V$ is not a finite-dimensional space the dual space $V^*$ may not be isomorphic to $V$, and not every functional $f: V\to F$ may be represented as some sort of scalar product of two elements of $V$. But in a finite-dimensional case $V^*$ is isomorphic to $V$ and your way of thinking about it is fine.