It isn't enough to have a large number of observations. You need multiple observations of each $i$ and $j$ for multiple time periods and doing trade with multiple countries. If, for example, if countries $k$ and $l$ only trade with each other, it wouldn't be possible to distinguish between $k$, $l$, and $lk$. In additions, you need multiple observations because you want to identify a time subscript on $i$ and $j$. If you have many countries, many periods, and many trade relationships, and the data are not sparse then this may not matter much.
You may find this paper of use:
Gravity or dummies? the limits of identification in gravity estimations (Hornok 2011)
This paper argues that identification of trade policy effects with a
gravity equation that includes country-time dummies to control for the
theoretical Multilateral Trade Resistances (MTR) is severely limited.
In most cases heterogeneous policy effects, i.e. more than one policy
dummies, cannot be identified separately, because the policy dummies
and the country-time dummies are perfectly collinear. Although a
single policy dummy can b e identified, the estimate may not b e
meaningful, because country-time dummies absorb to o much of the
useful variation of the data. Standard estimation techniques often do
not reveal these problems. The pap er demonstrates these arguments by
taking four typical research questions on the effect of a trade
policy, checking the
identifiability of the corresponding policy effect and deriving the
estimates. Empirical exercise on estimating the trade effects of EU
enlargement complements the analytical findings.
And Estimating the gravity model without gravity using panel data
(Westerlund and Wilhelmsson 2009)
"...$\alpha_{ij}$ which is unidentified in the fixed effects
formulation of the model. In order to identify $\alpha_{ij}$, a random
effects assumption is needed. But such assumptions are generally not
satisfied in practice..."
@ChinG, here is the basic gravity idea:
Newton's law of universal gravitation tells us that the force of gravity is
$$F = \frac{G M_1 M_2}{R^{2}_{1,2}}$$
which implies
$$f = g + m_1 + m_2 - 2 \cdot r_{1,2} $$
(where lowercase letters are the logs of upper case ones). In the trade equation, the force of gravity is replaced with the volume of trade. Usually, $M_1$ represents the economic size of economy one (say measured by $GDP_1$ and in your regression analogous to $I_t$), $M_2$ represents the economic size of economy two ($GDP_2$ and analogous to $J_t$), and $r_{1,2}$ is the geographic or effective economic distance between the two countries ($\alpha_{i,j}$). GDP isn't exactly the right measure because countries differ in fraction of the economy devoted to trad-able output. Think of it as analogous to the importance of that country in global trade (in an absolute and not relative sense).
