Seems like such polyhedra are called not necessarily closed (NNC) and are usually represented as closed polyhedra with additional dimension $\varepsilon$: every strict inequality $a_{i1}x_1 + \ldots + a_{in}x_n > c_i$ is replaced by $a_{i1}x_1 + \ldots + a_{in}x_n - \varepsilon \geq c_i$ and two additional inequalities $0 \leq \varepsilon \leq 1$ are added to the system. If we call this polyhedron $P'$, the desired polyhedron is the set $\{ (x_1,\ldots,x_n) \mid (x_1,\ldots,x_n,\varepsilon) \in P', \varepsilon > 0 \}$. Such a system of constraints can be converted to representation by generators.
Alternatively, they can be characterized directly by three sets of generators $R,P,C \in \mathbb{R}^n$, i.e. every point can be obtained as
$$\alpha_1r_1 + \ldots + \alpha_kr_k + \beta_1p_1 + \ldots + \beta_lp_l + \gamma_1c_1 + \ldots + \gamma_mp_m,$$
where $r_i \in R, p_i \in P, c_i \in C$ and $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}^+$ and $\sum_{i=1}^l \beta_i + \sum_{i=1}^m \gamma_i = 1$ and there is $1 \leq i \leq l$ such that $\beta_i \neq 0$. The trick here is that points in $C$ don't have to lie within the NNC polyhedron, but its closure, and whenever they appear in the sum, there must also be point from $P$ with nonzero coefficient. This representation can easily be converted to the one mentioned above.
References:
R. Bagnara, P. M. Hill, E. Zaffanella: A New Encoding of Not Necessarily Closed Convex Polyhedra
R. Bagnara, E. Ricci, E. Zaffanella, P. M. Hill: Possibly Not Closed Convex Polyhedra and the Parma Polyhedra Library
Seems like this stuff is used mostly in static analysis/verification and is not interesting to mathematicians, maybe I should have asked on cstheory instead? Any feedback regarding the question welcome, as well as corrections regarding my misuse of terminology/notation.