In axiomatic geometry, generally one does not give a definition - or in particular, the definition cannot be mathematical. Definitions like those in Euclid have no mathematical flavor, but are rather in human language, and meant to convey intuitive concepts and imagery so as to connect what the axioms are talking about to understandable experience and make their intent clear. The situation is not any different in the modern mathematical treatments - the difference is that our axioms are logically airtight for the manner in which modern mathematics is carried out and moreover that in a modern treatment one is up-front and honest that the terms like "straight line", etc. cannot be defined mathematically. They are what are called "primitive" concepts. Intuition, and intuitive understandings and concepts are, in my mind, VERY important, but they have to be guided with rigor. Rigor is the scaffolding, intuition is that in which it is dressed. The rigor cannot be truly understood without the intuition (and much problem solving is done on an intuitive level first), and intuition can lead one wrong if one does not also make sure it can be followed in rigor.
That said, if one takes the opposite approach of using analytic geometry, in which geometry is constructed out of the real number line using coordinate planes or spaces, then one can define the concept of a direction in space using calculus, and say a straight line is a curve that preserves a constant direction. The analytic approach, of course, is equivalent, but it admits of more generality in that it can be generalized to highly non-Euclidean, fluid spaces; it is the only approach in which things like Einstein's theory of general relativity can best be formulated.
(Historical note: This is, by the way, not saying Euclid was bad. The mathematics in his cultural area and time simply did not have the same methodology as now. His axioms are more like recipes for stating what sort of figures one is allowed to construct, but not so much recipes for how they are to behave and to interact. In particular, some scholars analyzing Euclid have suggested that his axioms, where they did more than prescribe constructions to make, were designed to convey what we would call the "metrical" - measuring or distance - aspect of Euclidean geometry, while the "topological" part - i.e. intersections - was to be handled by the use of diagrams and the diagram itself was an important reasoning tool. One can actually axiomatize diagrams separately to use them in a modern framework, but this approach is a bit wordier than it needs to be and most approaches just combine the topologic and metric axioms together. Though even with that, I still think a case can be made Euclid was truly incomplete, as some of his contemporaries or slightly later successors criticized his work on incompleteness grounds as well, for example, one criticism amounted to that he did not really know that a straight line could not meet with another in more than one point.)