You can define plane, angle, area, line, volume, curve, surface etc... first geometrically in $2$ and $3$ dimensions and then with the help of coordinates translate those terms in $\mathbb R^2$ and $\mathbb R^3$ in terms of equations satisfying some equalities or/and inequalities (line,plane,curve,surface) or in the terms of limits of sums (volume, area, length). All of geometrical intuition is helpful in $\mathbb R^2$ and $\mathbb R^3$ and, when you define distance from the origin of a point in $\mathbb R^2$ or $\mathbb R^3$ with the help of Pythagoras theorem then that distance is generalized to $\mathbb R^n$ as having exactly the same form as in $\mathbb R^2$ or $\mathbb R^3$ with only difference that it does not have $2$ or $3$ terms but $n$. Exactly the same is done with distance between any two points $x,y$. The properties of "distance function" are easily proven also in that general setting. Area and volume can be defined as set functions that have some properties we want them to have and then it is proved that integral of "nice" positive functions really is a good tool for defining area and volume in a sense that it has all the properties as area and volume defined as set functions. Angle is defined with a help of dot product and formula for dot product in $\mathbb R^n$ is straightforwardly generalized to have the same form as in $\mathbb R^2$ and $\mathbb R^3$.