How are parallel lines defined in n-dimensional Euclidean space?

Question

WolframMathWorld on parallel lines:

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect.

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Therefore, parallel lines in three-space lie in a single plane (Kern and Blank 1948, p. 9). Lines in three-space which are not parallel but do not intersect are called skew lines.

How are parallel lines commonly, or reasonably, defined for dimensions higher than three?

Answer 1

A good definition is:

Two straight lines are parallel if there is a plane that contains the two lines and the two lines have no common points.

This definition make sense also in $n-$dimensional spaces, with a suitable definition of what a plane is, as for an affine plane (because parallelism is an affine notion).

Answer 2

Let $P_1,P_2$ be distinct points on a line $l_1$ and $Q_1,Q_2$ be distinct points on a line $l_2$.

Then, $l_1$ and $l_2$ are parallel if and only if $\overrightarrow{P_1P_2} = k \cdot \overrightarrow{Q_1Q_2}$ for some $k \in \mathbb R$.