According to Wikipedia, the definition of Volume is given as "Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains."
The Wikipedia definition of volume is based on the ambient space being ordinary $3-$space. When we move to higher dimensions we may talk of $n-$volume to emphasize the fact that we are talking about that. A four dimensional brick has a $4-$volume that is computed by multiplying the four dimensions. Other four dimensional shapes can have their $4-$volumes computed by integration, just like more complicated shapes in $3$ dimensions.
It is quite possible to speak of the area of a cube-it is the surface area. A $4-$dimensional hypercube has eight cubical faces, you can speak of the total $3-$volume of those faces. You can also speak of the total area of the square faces of the eight cubes and so on. We don't have everyday words for the content of things in higher dimensions because most people don't worry about them.