When we take the wedge product of 3 vectors result is, $$ {\displaystyle \mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w} =(u_{1}v_{2}w_{3}+u_{2}v_{3}w_{1}+u_{3}v_{1}w_{2}-u_{1}v_{3}w_{2}-u_{2}v_{1}w_{3}-u_{3}v_{2}w_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3})} $$
Scalar part is the volume of the parallelepiped and trivector part is the "direction" of that volume. But what's that mean at all?
I've no problem with directional lines such as vectors, we're using them in many fields and they're pretty useful.
But in the case of directional volumes it's really hard to imagine.
Are they practically used in any field? And what do they represent in real world?
Trivectors in $\mathbf{R}^3$ are perhaps not that interesting, just like a flatlander living in $\mathbf{R}^2$ would perhaps struggle to see what the point with bivectors is. After all, in $\mathbf{R}^2$ any wedge product $\mathbf{u} \wedge \mathbf{v}$ just equals a constant times the bivector $\mathbf{e}_1 \wedge \mathbf{e}_2$ (and that constant is of course plus or minus the area of the parallelogram whose edges are given by the vectors $\mathbf{u}$ and $\mathbf{v}$).
But in $\mathbf{R}^3$, the wedge product $\mathbf{u} \wedge \mathbf{v}$ is more interesting, since it somehow represents a “two-dimensional direction”, the subspace (or plane) spanned by $\mathbf{u}$ and $\mathbf{v}$ (with an orientation). And there are infinitely many such two-dimensional subspaces here, not just one like in $\mathbf{R}^2$.
Similarly, in higher dimensions, like in $\mathbf{R}^4$ to begin with, a triple wedge product $\mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w}$ not only has a magnitude (the three-dimensional volume given by the three vectors), it also represents a “three-dimensional direction”, namely the (oriented) subspace spanned by $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$. And there are infinitely many such three-dimensional subspaces here, not just one like in $\mathbf{R}^3$.
That said, bivectors are actually conceptually useful already in $\mathbf{R}^2$. For example, one can use a complex number $a+ib$ to rotate another complex number $c+id$ (by multiplication), but it's conceptually better to use the bivector $a+b \mathbf{I}$ (where $\mathbf{I}=\mathbf{e}_1 \wedge \mathbf{e}_2$) to rotate the vector $c \mathbf{e}_1 + d \mathbf{e}_2$ (by conjugation), since that's how rotations work in higher dimensions, where the algebraic objects representing the rotations are of a different kind than the vectors being rotated. And it's also nice to have trivectors in $\mathbf{R}^3$, since they have a natural role to play in the Clifford algebra.