Generalizing Lami's theorem In statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem

$$\frac{F_1}{\sin{\alpha}}=\frac{F_2}{\sin{\beta}}=\frac{F_3}{\sin{\gamma}}.\tag{1}$$

where $F_1$, $F_2$ and $F_3$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors which keep the object in static equilibrium, and $\alpha$, $\beta$ and $\gamma$ are the angles directly opposite to the vectors (see Figure 1).

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy. Its proof is essentially based on the law of sines.
On the Internet there are hundreds of static equilibrium problems where they apply Lami's theorem to a three-force system, see for instance Dubey - Engineering Mechanics: Statics and Dynamics, section 3.10. Although Dubey's book is recent (2013), there is not a single equilibrium problem based on a four-force system. Coincidentally, the author of this note has come across questions on the Internet questioning the possibility of applying Lami's theorem for more than three forces. In this note we give a generalization of Lami's theorem for four forces.
Theorem 1 (Generalization). If four coplanar, concurrent and non-collinear forces act upon an object, and the object remains in static equilibrium, then


$$AD\sin{\alpha'}+BC\sin{\gamma'}=AB\sin{\beta'}+CD\sin{\delta'}.\tag{2}$$

where $A$, $B$, $C$ and $D$ are the magnitudes of the four vectors and $\alpha'$, $\beta'$, $\gamma'$ and $\delta'$ are the angles between them (see Figure 2).
Proof. Consider the quadrilateral formed by the four vectors in such a manner that the head of one touches the tail of another (see Figure 3) and denote $\Delta$ its area. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the interior angles of the quadrilateral, then its area can be written as

$$\Delta=\frac12AD\sin{\alpha}+\frac12BC\sin{\gamma}=\frac12AB\sin{\beta}+\frac12CD\sin{\delta}\tag{3}$$
and as $\sin{\alpha'}=\sin{(\pi-\alpha)}=\sin{\alpha}$, and similarly for $\beta'$, $\gamma'$ and $\delta'$, the relation in $(2)$ follows. $\square$
Theorem 1 is a generalization in the sense that if one of the vectors vanishes, the relation we obtain is that of Lami's theorem. Indeed, for instance suppose $C=0$, then the relation $(2)$ reduces to
$$D\sin{\alpha'}=B\sin{\beta'},\tag{4}$$
which is Lami's theorem.
Remark. A generalization of Lami's theorem is given by H.Shekhar. However, this generalization is different since it only considers cyclic polygons with an odd number of sides.
Question: Can Theorem 1 generalize to higher dimensions?
 A: I'll describe the 3D version of Lami's theorem that I think you want.  It's not original, but I've tried to adapt it to your requirements as I understand them.
It uses a 3D generalization of $\sin()$ that we'll call $\sin_{3D}()$.
$\sin(\alpha)$ assigns a value to a planar angle $\alpha$.  $\sin_{3D}(\delta)$ assigns a value to a 3D trihedral angle $\delta$.
Given four vectors $F_i$ that sum to $0$, we'll show that there are $\delta_i$ (dependent on $F_{j\ne i}$) such that all $i,j$
$$\dfrac{|F_i|}{\sin_{3D}(\delta_i)}=\dfrac{|F_j|}{\sin_{3D}(\delta_j)}.\tag{1}$$
Further, $\sin_{3D}$ can be expressed as a product of $\sin()$ terms.
(please note that while the OP defines $F_i$ as a magnitude, it's easier here to talk about it as a vector).
Just as for the 2D case, the proof is essentially the 3D Law of Sines, which is developed in Allendoerfer, Generalizations of Theorems about Triangles (also available here).  In that paper, the 3D Law of Sines concerns the faces of a tetrahedron and the $\sin_{3D}()$s of their opposite trihedral angles. (there, $\sin_{3D}()$ is referred to as $\operatorname{G-sin}()$).  See Theorem 4, pg 257.
We haven't defined trihedral angles yet.  A trihedral angle is defined by three vectors $v_1,v_2,v_3$, and $\delta=\delta(v_1,v_2,v_3)$ can be thought of as the space bounded by the three planes defined by pairs of these vectors. Let $u_1,u_2,u_3$ be the unit vectors corresponding to $v_1,v_2,v_3$.  Then $\sin_{3D}(\delta)$ is defined as the volume of the parallelopiped $P$ with edges $u_1,u_2,u_3$.  It's well known that the volume of $P$ is the triple product of $u_1,u_2,u_3$, e.g. $u_1\cdot u_2\times u_3$. Further, if $\alpha$ is the angle between $u_2$ and $u_3$, and $\beta$ is the dihedral angle between $u_1$ and the plane defined by $u_2,u_3$, then the volume of $P$ reduces to $\sin(\alpha)\sin(\beta).$
So, for example, a term in $(1)$ would be written as something like
$$ 
\dfrac{|F_1|}{\sin_{3D}(\delta(F_2,F_3,F_4))}=\dfrac{|F_1|}{\sin(\alpha_1)\sin(\beta_1)},\tag{2}
$$
where $\alpha_1=\angle(F_3,F_4)$ and $\beta_1$ is a dihedral angle as described above.
We haven't explained how $(1)$ or $(2)$ is essentially the 3D Law of Sines, which concerns the areas of the faces of a tetrahedron.  The bridge is that a face can be interpreted as a vector pointing in the direction of its normal with magnitude its area.  Via Minkowski's Problem you can go back and forth between tetrahedra and sets of four vectors that sum to $0$.
The above is just a sketch, and you'll have to piece it together but hopefully (2) is the type of generalization you wanted.
Allendoerfer mentions in the paper that the higher dimensional Law of Sines goes back to Grassmann.
Speaking of exterior algebra, here's a treatment that uses wedge products and Hodge stars but is much the same as the above.
