No, skew lines cannot be in the same plane, since they do not have a point on common. Two lines intersect if they lie in a common plane, and by definition, these intersecting lines are not skew lines.
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Parallel Lines - Skew lines are lines that do not intersect and do not lie in the same plane. If two lines lie in the same plane they they must either intersect or they are parallel.
parallel lines If they are not on the same plane and never intersect they are skew
If they are not parallel and do not intersect then they would not be in the same plane (not possible), but they would be called skew lines.
skew
If the 2 lines lie in the same plane, and they are not parallel, then they will intersect at some point. If the 2 lines are skew lines, then they are not in the same plane, and they will not intersect (but they are Not Parallel)